Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting

Considering the question: how non-linear may a non-linear operator be in order to extend the linear regularization theory, we introduce the class of dilinear mappings, which covers linear, bilinear, and quadratic operators between Banach spaces. The corresponding dilinear inverse problems cover blind deconvolution, deautoconvolution, parallel imaging in MRI, and the phase retrieval problem. Based on the universal property of the tensor product, the central idea is here to lift the non-linear mappings to linear representatives on a suitable topological tensor space. At the same time, we extend the class of usually convex regularization functionals to the class of diconvex functionals, which are likewise defined by a tensorial lifting. Generalizing the concepts of subgradients and Bregman distances from convex analysis to the new framework, we analyse the novel class of dilinear inverse problems and establish convergence rates under similar conditions than in the linear setting. Considering the deautoconvolution problem as specific application, we derive satisfiable source conditions and validate the theoretical convergence rates numerically.


Introduction
Nowadays the theory of inverse problems has become one of the central mathematical approaches to solve recovery problems in medicine, engineering, and life sciences. Some of the main applications are computed tomography (CT), magnetic resonance imaging (MRI), and deconvolution problems in microscopy, see for instance [BB , MS , Ram , SW , Uhl , Uhl ] being recent monographs as well as many other publications.
The beginnings of the modern regularization theory for ill-posed problems are tracing back to the pioneering works of A. N. T [Tik a, Tik b]. Between then and now, the theory has been heavily extended and covers linear and non-linear for-mulations in the H space as well as in the B space setting. In order to name at least a few of the numerous monographs, we refer to [BG , EHN , Hof , Lou , LP , Mor , SKHK , TA , TLY ]. Due to the enormous relevance of the research topic, the published literature embraces many further monographs as well as a vast number of research articles.
Especially for linear problem formulations, the analytical framework is highly developed and allows the general treatment of inverse problems with continuous operators, see for instance [EHN , MS , SKHK ] and references therein. In addition to the sophisticated analysis, efficient numerical implementations of the solution schemes are available for practitioners [EHN , Sch ]. The interaction between analysis and numerics are one reason for the great, interdisciplinary success of the linear regularization theory for ill-posed inverse problems.
If the linear operator in the problem formulation is replaced by a non-linear operator, the situation changes dramatically. Depending on the operator, there are several regularization approaches with different benefits and drawbacks [EHN , Gra , HKPS , SGG + ]. One standard approach to regularize non-linear operators is to introduce suitable non-linearity conditions and to restrict the set of considered operators. These constraints are mostly based on properties of the remainder of the first-order T expansion [BO , EKN , HKPS , RS ]. In an abstract way, this approach allows the generalization of well-understood linear results by controlling the deviation from the linear setting. Unfortunately, the validation of the required assumptions for a specific non-linear operator is a non-trivial task.
In order to extend the linear theory to the non-linear domain further, our idea is to introduce a class of operators that covers many interesting applications for practitioners and, at the same time, allows a general treatment of the corresponding inverse problems. More precisely, we introduce the class of dilinear operators that embraces linear, bilinear, and quadratic mappings between B spaces. Consequently, our novel class of dilinear inverse problems covers formulations arising in imaging and physics [SGG + ] like blind deconvolution [BS , JR ], deautoconvolution [GH , FH , GHB + , ABHS ], parallel imaging in MRI [BBM + ], or phase retrieval [DF , Mil , SSD + ].
The central idea behind the class of dilinear operators is the universal property of the topological tensor product, which enables us to lift a continuous but non-linear mapping to a linear operator. Owing to the lifting, we get immediate access to the linear regularization theory. On the downside, a simple lifting of the non-linear inverse problem causes an additional non-convex rank-one constraint, which is similarly challenging to handle than the original non-linear problem. For this reason, most results of the linear regularization theory are not applicable for the lifted problem and cannot be transferred to the original (unlifted) inverse problem. In order to overcame this issues, we use the tensorial lifting indirectly and generalize the required concepts from convex analysis to the new framework.
The recent literature already contains some further ideas to handle inverse problems arising from bilinear or quadratic operators. For instance, each quadratic mapping between separable H spaces may be factorized into a linear operator and a strong quadratic isometry so that the corresponding inverse problem can be decomposed into a possibly ill-posed linear and a well-posed quadratic part, see [Fle ]. In order to determine a solution, one can now apply a two-step method. Firstly, the ill-posed linear part is solved by a linear regularization method. Secondly, the well-posed quadratic problem is solved by projection onto the manifold of symmetric rank-one tensors. The main drawback of this approach is that the solution of the linear part has not to lie in the range of the well-posed quadratic operator such that the second step may corrupt the obtained solution. This issue does not occur if the forward operator of the linear part is injective, which is generally not true for quadratic inverse problems.
Besides the forward operator, one can also generalize the used regularization from usually convex to non-convex functionals. An abstract analysis of non-convex regularization methods for bounded linear operators between H spaces has been introduced in [Gra ], where the definition of the subdifferential and the B distance have been extended with respect to an arbitrary set of functions. On the basis of a variational source condition, one can further obtain convergence rates for these non-convex regularization methods. Similarly to [Gra ], we employ non-convex regularizationshowever -with the tensorial lifting in mind.
As mentioned above, the deautoconvolution problem is one specific instance of a dilinear inverse problem [GHB + , ABHS ]. Although the unregularized problem can have two different solutions at the most, the deautoconvolution problem is everywhere locally ill posed [GH , FH , Ger ]. Nevertheless, with an appropriate regularization, very accurate numerical solutions can be obtained [CL , ABHS ]. Establishing theoretical convergence rates for the applied regularization is, unfortunately, very challenging since most conditions for the non-linear theory are not fulfilled [BH , ABHS ]. For more specific classes of true solutions, for instance, the class of all trigonometric polynomials or some subset of S spaces, however, the regularized solutions converge to the true solution with a provable rate, see [BFH ] or [Jan , DL ] respectively. Applying our novel regularization theory, we establish convergence rates under a source-wise representation of the subdifferential of the regularization functional. In other words, we generalize the classical range source condition in a specific manner fitting the necessities of dilinear inverse problems.
In this paper, we show that the essential results of the classical regularization theory with bounded linear operators and convex regularization functionals may be extended to bilinear and quadratic forward operators. At the same time, we allow the regularization functional to be non-convex in a manner being comparable with the non-linearity of the considered operator.
Since our analysis is mainly based on the properties of the topological tensor product [DF , Rya ], we firstly give a brief survey of tensor spaces and the tensorial lifting in Section . Our main foci are here the different interpretations of a specific tensor. Further, we introduce the set of dilinear operators and show that each dilinear operator may be uniquely lifted to a linear operator. Analogously, in Section , the class of diconvex functionals is defined through a convex lifting. In order to study diconvex regularization methods, we generalize the usually convex subdifferential and B distance with respect to dilinear mappings. Further, we derive sum and chain rules for the new dilinear subdifferential calculus.
Our main results about dilinear -bilinear and quadratic -inverse problems are presented in Section . Under suitable assumptions being comparable to the assumptions in the classical linear theory, dilinear inverse problems are well posed, stable, and consistent. Using a source-wise representation as in the classical range condition, we obtain convergence rates for the data fidelity term and the B distance between the true solution of the undisturbed and the regularized problem.
In Section , we apply our non-convex regularization theory for dilinear inverse problems to the deautoconvolution problem and study the required assumptions in more detail. Further, we reformulate the source-wise representation and obtain an equivalent source condition, which allows us to construct suitable true solutions. With the numerical experiments in Section , we verify the derived rates for the deautoconvolution numerically and give some examples of possible signals fulfilling the source condition.

. Tensor products and dilinear mappings
The calculus of tensors has been invented more than a century ago. Since then tensors have been extensively studied from an algebraic and topological point of view, see for instance [DF , DFS , Lic , Rom , Rya ] and references therein. One of the most remarkable results in tensor analysis, which is the starting point for our study of nonlinear inverse problems, is the universal property of the tensor product that allows us to lift a bilinear mapping to a linear one. In order to get some intuition about tensor calculus on B spaces and about the different interpretations of a specific tensor, we briefly survey the required central ideas about tensor products and adapt the lifting approach to the class of dilinear forward operators.
The tensor product of two real vector spaces V and V denoted by V ⊗ V can be constructed in various ways. Following the presentations of R [Rya ] and D et al. [DFS ], we initially define a tensor as a linear operator acting on bilinear forms. For this purpose, we recall that a mapping from V ×V into the vector space W is bilinear if it is linear with respect to each variable. The corresponding vector space of all bilinear mappings is denoted by B(V × V ,W ). In the special case W = R, we simply write B(V × V ) for the vector space of all bilinear forms. Given two elements u ∈ V and ∈ V , we now define the tensor u ⊗ as the linear functional that evaluates a bilinear form at the point (u, ). The tensor u ⊗ thus acts on a bilinear form On the basis of this construction, the tensor product V ⊗ V of the real vector spaces V and V now consists of all finite linear combinations w = N n= λ n u n ⊗ n with u n ∈ V , n ∈ V , λ n ∈ R, and N ∈ N. Thus, the tensor product V ⊗ V is the subspace of the algebraic dual B(V × V ) # spanned by the rank-one tensors u ⊗ with u ∈ V and ∈ V . Using the definition of a tensor as linear functional on B(V × V ), one can easily verify that the mapping (u, ) → u ⊗ itself is bilinear. In other words, we have One of the most central concepts behind the calculus of tensors and most crucial for our analysis of non-linear inverse problems is the linearization of bilinear mappings. Given a bilinear mapping A: V ×V → W , we define the linear mappingȂ : V ⊗V → W byȂ( N n= u n ⊗ n ) = N n= A(u n , n ). One can show that the mappingȂ is well defined and is the only linear mapping in L(V ⊗ V ,W ) such that A(u, ) =Ȃ(u ⊗ ) for all u ∈ V and ∈ V , see for instance [Rya ].
Proposition . (Li ing of bilinear mappings). Let A: V × V → W be a bilinear mapping, where V , V , and W denote real vector spaces. Then there exists a unique linear mappingȂ : Besides the definition of a tensor as linear functional on B(V × V ), we can interpret a tensor itself as a bilinear form. For this purpose, we associate to each u ∈ V and ∈ V the bilinear form B u ⊗ : Moreover, the mapping (u, ) → B u ⊗ is also bilinear, which implies that there is a linear mapping from V ⊗ V into B(V # × V # ). Since this mapping is injective, see [Rya ], each tensor w in V ⊗ V corresponds uniquely to a bilinear mapping, and hence Further, the tensor product V ⊗ V can be seen as vector space of linear mappings [DFS , Rya ]. For this, we observe that each bilinear mapping A ∈ B(V ×V ) generates the linear mappings by fixing one of the components. In this context, we may write On the basis of this idea, each tensor w = N n= λ n u n ⊗ n in V ⊗ V generates the linear mappings . If one of the real vector spaces V or V is already a dual space, we have a natural embedding into the smaller linear function spaces There are several approaches to define an appropriate norm on the tensor product V ⊗V . Here we employ the projective norm, which allow us to lift each bounded bilinear operator to a bounded linear operator on the tensor product. To define the projective norm, we assume that the real vector spaces V and V above are already equipped with an appropriate norm. More precisely, we replace the arbitrary real vector spaces V and V by two B spaces X and X . The projective norm is now defined in the following manner, see for instance [DF , Rya ].
Definition . (Projective norm). Let X and X be real B spaces. The projective norm || · || π on the tensor product X ⊗ X is defined by where the infimum is taken over all finite representations of w ∈ X ⊗ X .
The projective norm on X ⊗ X belongs to the reasonable crossnorms, which means that ||u ⊗ || π = ||u || || || for all u ∈ X and ∈ X , see [DFS , Rya ]. Based on the projective norm, we obtain the projective tensor product X ⊗ π X .
Definition . (Projective tensor product). Let X and X be real B spaces. The projective tensor product X ⊗ π X of X and X is the completion of the tensor product X ⊗ X with respect to the projective norm || · || π . Figuratively, we complete the tensor product X ⊗ X consisting of all finite-rank tensors by the infinite-rank tensors w with Similarly as above, if one of the B spaces X or X is a dual space, the projective tensor product can be embedded into the space of bounded linear operators. More precisely, we have X * ⊗ π X ⊂ L(X , X ) and X ⊗ π X * ⊂ L(X , X ), see for instance [Won ]. In the H space setting, the projective tensor product H ⊗ π H of the H spaces H and H corresponds to the trace class operators, and the projective norm is given by ||w || π = ∞ n= σ n (w) for all w ∈ H ⊗ π H , where σ n (w) denotes the nth singular value of w, see [Wer ].
The main benefit of equipping the tensor product with the projective norm is that Proposition . remains valid for bounded bilinear and linear operators. For this, we recall that a bilinear operator A: X × X → Y is bounded if there exists a real constant C such that ||A(u, ) | | ≤ C ||u || || || for all (u, ) in X × X . Following the notation in In the special case Y = R, we again write B(X × X ). More precisely, the lifting of bounded bilinear operators can be stated in the following form, see for instance [Rya ].
Proposition . (Li ing of bounded bilinear mappings). Let A: X × X → Y be a bounded bilinear operator, where X , X , and Y denote real B spaces. Then there exists a unique bounded linear operatorȂ : The lifting of a bilinear operator in Proposition . is also called the universal property of the projective tensor product. Vice versa, each bounded linear mappingȂ: X ⊗ π X → Y uniquely defines a bounded bilinear mapping A by A(u, ) =Ȃ(u ⊗ ), which gives the canonical identification Consequently, the topological dual of the projective tensor product X ⊗ π X is the space B(X ×X ) of bounded bilinear forms, where a specific bounded bilinear mapping A: X × X → R acts on an arbitrary tensor w = ∞ n= λ n u n ⊗ n by see for instance [Rya ]. In order to define the novel class of dilinear operators, we restrict the projective tensor product to the subspace of symmetric tensors. Assuming that X is a real B space, we call a tensor w ∈ X ⊗ π X symmetric if and only if w = w T , where the transpose of w = Hence, the dilinear mappings are the restrictions of the linear operators from X × (X ⊗ π,sym X ) into Y to the diagonal {(u, u ⊗ u) : u ∈ X }. Since the representativeK acts on a C ian product, we can always find two linear mappingsȂ : X → Y and B : X ⊗ π,sym X → Y so thatK(u, w) =Ȃ(u) +B(w). A dilinear mapping K is bounded if the representative linear mappingK is bounded. It is worth to mention that the repres-entativeK of a bounded dilinear operator K is uniquely defined.
Lemma . (Li ing of bounded dilinear mappings). Let K : X → Y be a bounded dilinear mapping, where X and Y denotes real B spaces. Then the (bounded) repres-entativeK is unique.
Proof. Let us suppose that there exist two bounded representativesK andK for the bounded dilinear mapping K. Since both representativesK ℓ can be written asK for all u in X . By replacing u by tu with t ≥ and ||u || = , this identity is equivalent to which already impliesȂ =Ȃ . Due to the continuity and linearity ofB ℓ , the mappings B ℓ coincide on the symmetric subspace X ⊗ π,sym X , which yields the assertion.
Remark . (Unique dilinear li ing). Because of the uniqueness in Lemma . , the breve· henceforth denotes the unique lifting of a bounded dilinear mapping.
Example . (Linear mappings). One of the easiest examples of a dilinear operator are the linear operators A: X → Y with the representativeȂ(u, u ⊗ u) = A(u). Consequently, the dilinear operators can be seen as a generalization of linear mappings.
Example . ( adratic mappings). A further example of dilinear mappings are the quadratic mappings. For this, we recall that a mapping Q : X → Y is quadratic if there exists a bounded symmetric bilinear mapping A: X ×X → Y such that Q(u) = A(u, u) for all u in X . Since each bilinear operator is uniquely liftable to the tensor product X ⊗ π X by Proposition . , the representative of Q is just given byQ(u, u ⊗ u) =Ȃ(u ⊗ u), wherȇ A is the restriction of the lifting of A to the subspace X ⊗ π,sym X .
Example . (Bilinear mappings). Finally, the dilinear mappings also cover the class of bounded bilinear operators. For this, we replace the B space X by the C ian product X ×X , where X and X are arbitrary real B spaces. Given a bounded bilinear operator A: X × X → Y , we define the symmetric bilinear mapping B : (X × X ) × (X × X ) → Y by B((u , ), (u , )) = / A(u , ) + / A(u , ). Using the lifting B of B, we obtain the representativeȂ((u, ), (u, ) ⊗ (u, )) =B((u, ) ⊗ (u, )) for all (u, ) in X × X .

. Generalized subgradient
One of the drawbacks of the dilinear operators in Definition . is that dilinear mappings K : X → R do not have to be convex. Hence, the application of the usual subgradient to dilinear operators is limited. To surmount this issue, we generalize the concept of convexity and of the usual subgradient, see for instance [BC , ET , Roc ], to our setting. In the following, we denote the real numbers extended by +∞ and −∞ by R. For a mapping F between a real B space and the extended real numbers R, the effective domain is the section The mapping F : X → R is proper if it is never −∞ and not everywhere +∞.
Definition . (Diconvex mappings). Let X be real B spaces. A mapping Since each proper, convex mapping F : X → R may be represented by the convex mappingF (u, u ⊗ u) = F (u) on X × (X ⊗ π,sym X ), we can view the diconvex mappings as a generalization of the set of proper, convex mappings. The central notion behind this definition is that each dilinear functional is by definition also diconvex. However, differently from the dilinear operators, the representativeF of a diconvex mapping F does not have to be unique. As we will see, one sufficient condition for diconvexity is the existence of a continuous, diaffine minorant A, where diaffine means that there exists a continuous, affine mappingȂ : X ×(X ⊗ π,sym X ) → Y such that A(u) =Ȃ(u, u ⊗u) for all u in X . In this context, a continuous mappingȂ is affine if and only ifȂ(u, w) ≔ T (u, w)+t for a bounded linear operator T : X × (X ⊗ π,sym X ) → R and a constant t ∈ R.
In order to prove this assertion, we will exploit that each vector (u, u ⊗u) is an extreme point of the convex hull of the diagonal {(u, u ⊗ u) : u ∈ X }, which means that (u, u ⊗ u) cannot be written as a non-trivial convex combination of other points, see [Roc ].
Lemma . (Extreme points of the convex hull of the diagonal). Let X be a real Proof. For an element (u, u ⊗ u) with u ∈ X , we consider an arbitrary convex combination ], and N n= α n = . Applying the linear functionals (ϕ, ⊗ ) and ( , ϕ ⊗ ϕ) with ϕ ∈ X * of the dual space X * × (X ⊗ π,sym X ) * , we get the identity N n= α n ϕ, u n = ϕ, u = ϕ ⊗ ϕ, u ⊗ u = N n= α n ϕ, u n .
Due to the strict convexity of the square, this equation can only hold if ϕ, u n = ϕ, u for every n between and N , and for every ϕ ∈ X * . Consequently, all u n coincide with u, which shows that the considered convex combination is trivial, and that (u, u ⊗ u) is an extreme point.
With the knowledge that the diagonal {(u, u ⊗ u) : u ∈ X } contains only extreme points of its convex hull, we may now give an sufficient condition for a mapping being diconvex.
Proposition . . Let X be a real B space. If the mapping F : X → R has a continuous, diaffine minorant, then F is diconvex.
Proof. If F has a continuous, diaffine minorant G with representativeG, we can construct a representativeF by else.
Since we firstly restrict the convex mappingG to the convex set {(u, u ⊗ u) : u ∈ X } and secondly increase the function values of the extreme points on the diagonal, the constructed mappingF is convex. Obviously, the functionalF is also proper and thus a valid representative.
Remark . . If the B space X in Proposition . is finite-dimensional, then the reverse implication is also true. To validate this assertion, we restrict a given proper, convex representativeF to the convex hull of {(u, u ⊗ u) : u ∈ X }, which here means that we setF ( , w) = +∞ for ( , w) outside the convex hull. Due to the fact that the relative interior of a convex set is non-empty in finite dimensions, see [Roc , Theorem . ], there exists a point ( , w) where the classical subdifferential ∂F ( , w) of the proper, convex representativeF is non-empty, see for instance [Roc , Theorem . ]. In other words, we find a dual element (ξ , Ξ) ∈ X * × (X ⊗ π,sym X ) * such that for all ′ ∈ X and w ′ ∈ X ⊗ π,sym X . Obviously, the functionalȂ given by defines a continuous, affine minorant ofF , and the restriction A(u) ≔Ȃ(u, u ⊗ u) thus a diaffine minorant A of F .
Remark . . Looking back at the proof of Proposition . , we can directly answer the question: why does the representative of a diconvex mapping has to be proper? If we would allow inproper representatives as well, every mapping F : X → R would be diconvex with the convex but inproper representativȇ so the diconvex mappings would simply embrace all possible mappings between the B space X and R. The condition that the representative is proper will be needed at several points for the generalized subdifferential calculus and the developed regularization theory.
After this preliminary considerations, we now generalize the classical subgradient and subdifferential to the class of (proper) diconvex mappings.
for all in X . The union of all dilinear subgradients of F at u is the dilinear subdifferential ∂ β F (u). If no dilinear subgradient exists, the dilinear subdifferential is empty.
If the mapping F is convex, then the dilinear subdifferential obviously contains the usual subdifferential of F . More precisely, we have ∂ β F (u) ⊃ ∂F (u) × { }. Where the usual subgradient consists of all linear functionals entirely lying below the mapping F , the dilinear subgradient consists of all dilinear mappings below F . In this context, the dilinear subdifferential can be interpreted as the W -subdifferential introduced in [Gra ] with respect to the family of dilinear functionals whereas the bilinear part Ξ here does not have to be negative semi-definite. If we think at the one-dimensional case F : R → R, the dilinear subdifferential embraces all parabolae beneath F at a certain point u. Since each diconvex mapping F has at least one convex representativeF , we next investigate how the representativeF may be used to compute the dilinear subdifferential ∂ β F (u).
Definition . (Representative subgradient). Let F : X → R be a diconvex mapping on the real B space X with representativeF : X × (X ⊗ π,sym X ) → R. The dual element (ξ , Ξ) ∈ X * × (X ⊗ π,sym X ) * is a representative subgradient of F at u with respect toF if (ξ , Ξ) is a subgradient of the representativeF at (u, u ⊗ u). The union of all representative subgradients of F at u is the representative subdifferential∂F (u) with respect toF .
Since the representativeF of a diconvex mapping F may not be unique, the representative subgradient usually depends on the choice of the mappingF . Nevertheless, a representative subgradient is as well a dilinear subgradient.
Lemma . (Inclusion of subdifferentials). Let F : X → R be a diconvex mapping on the real B space X with representativeF . Then the representative and dilinear subdifferential are related by∂ Proof. Since each representative subgradient (ξ , Ξ) fulfils for all ( , w) in X × (X ⊗ π,sym X ) and thus especially for ( , w) = ( , ⊗ ), the asserted inclusion follows.
In view of the fact that the representativeF of a diconvex mapping F is not unique, the question arises whether there exists a certain representativeF such that the representative subdifferential coincides with the dilinear subdifferential. Indeed, we can always construct an appropriate representative by considering the convexification of F on X ×(X ⊗ π,sym X ). In this context, the convexification conv G of an arbitrary functional G : X → R on the B space X is the greatest convex function majorized by G and can be determined by where the infimum is taken over all convex representations u = N n= α n u n with n ∈ N, u n ∈ X , and α n ∈ [ , ] so that N n= α n = , see for instance [Roc ]. For a diconvex functional F : X → R, we now consider the convexification of Obviously, the mapping conv F ⊗ as supremum of all convex functionals majorized by F ⊗ is a valid representative of F since there exists at least one convex representativeF with F (u) =F (u, u ⊗ u) for all u in X . Further, the convexification conv F ⊗ is also proper since the representativeF has to be proper.
Theorem . (Equality of subdifferentials). Let F : X → R be a diconvex mapping on the real B space X . Then the representative subdifferential with respect to conv F ⊗ and the dilinear subdifferential coincide, i.e.
Taking the infimum over all convex combinations Similarly to the classical (linear) subdifferential, the dilinear subdifferential of a sum F + G contains the sum of the single dilinear subdifferentials of F and G.
Proposition . (Sum rule). Let F : X → R and G : X → R be diconvex mappings on the real B space X . Then the dilinear subdifferential of F + G and the dilinear subdifferentials of F and G are related by Proof. This is an immediate consequence of Definition . . Differently from the classical (linear) subdifferential, we cannot transfer the chain rule to the dilinear/diconvex setting. The main reason is that the composition of a diconvex mapping F and a dilinear mapping K has not to be diconvex, since a representative of F • K cannot simply be constructed by composing the representatives of F and K. Therefore, the chain rule can only be transferred partly. For an arbitrary bounded linear operator K : X → Y , we recall that there exists a unique bounded linear operator [Rya ]. In the following, the restriction of the lifted operator K ⊗ π K to the symmetric subspace is denoted by K ⊗ π,sym K : of the mappings K and K ⊗ π,sym K is defined by Proposition . (Chain rule for linear operators). Let K : X → Y be a bounded linear mapping and F : Y → R be a diconvex mapping on the real B spaces X and Y . Then the dilinear subdifferential of F • K is related to the dilinear subdifferential of F by Proof. Firstly, we notice that the functional F • K is diconvex with the representativȇ F • (K × (K ⊗ π,sym K)). Next, let us assume (ξ , Ξ) ∈ ∂ β F (K(u)), which is equivalent to Proposition . (Chain rule for convex functionals). Let K : X → Y be a bounded dilinear mapping and F : Y → R be a convex mapping on the real B spaces X and Y . Then the dilinear subdifferential of F • K is related to the linear subdifferential of F by Proof. The functional F • K is diconvex with convex representative F •K, since the lifted operatorK is linear. Next, we consider a linear subgradient ξ of F at K(u), which means Since the representative subdifferential is based on the classical subdifferential on the lifted space, the classical sum and chain rules obviously remain valid whenever the representatives fulfil the necessary conditions. For instance, one functional of the sum or the outer functional of the composition is continuous at some point. At least, for finite-dimensional B spaces, the representative of a diconvex functional, which is finite on some open set, is continuous on the interior of its effective domain. The central idea to prove this conjecture is that the manifold of rank-one tensors is curved in a way such that the convex hull of each open set of the manifold contains an inner point with respect to the surrounding tensor product. In the following, we denote by B ϵ the closed ϵ-ball around zero.
Lemma . (Local convex hull of rank-one tensors). Let u be a point of the real finite-dimensional B space X . The interior of the convex hull of the set is not empty for every ϵ > .
Proof. To determine a point in the interior, we construct a suitable simplex by using a normalized basis (e n ) N n= of the B space X . Obviously, the convex hull contains the points (u + ϵe n , (u + ϵe n ) ⊗ (u + ϵe n )). Viewing the point (u, u ⊗ u) as new origin of X × (X ⊗ π,sym X ), we obtain the vectors where the first components again form a basis of the first component space X . Next, we consider the convex combination of (u + ϵe n , (u + ϵe n ) ⊗ (u + ϵe n )) and (u − ϵe n , (u − ϵe n ) ⊗ (u − ϵe n )) with weights / . In this manner, we obtain (u + ϵe n , (u + ϵe n ) ⊗ (u + ϵe n ))

( )
Since the vectors (e n ⊗ e m ) N n,m= form a basis of the tensor product X ⊗ π X , see [Rya ], the vectors in ( ) and ( ) span the second component space X ⊗ π,sym X . Thus, the vectors ( -) form a maximal set of independent vectors, and the convex hull of them cannot be contained in a true subspace of X × (X ⊗ π,sym X ). Consequently, the simplex spanned by the vectors ( -) and zero contains an inner point. Since the constructed simplex shifted by (u, u ⊗ u) is contained in the convex hull of ( ), the assertion is established.
Unfortunately, Lemma . does not remain valid for infinite-dimensional B spaces. For example, if the B space X has a normalized S basis (e n ) n ∈N , we can explicitly construct a vector not contained in the convex hull of ( ) but arbitrarily near to a given element of the convex hull. For this, we notice that the tensor product X ⊗ π X possesses the normalized S basis (e n ⊗ e m ) n,m ∈N with respect to the square ordering, see [Rya ], and that the coordinates of an arbitrary rank-one tensor u ⊗ u = (n,m)∈N a nm (e n ⊗ e m ) with u = n ∈N u n e n are given by a nm = u n u m . Consequently, the coordinates a nn on the diagonal have to be non-negative. This implies that the convex hull of ( ) only contains tensors with non-negative diagonal. Now, let ( , w) be an arbitrary element of the convex hull of ( ). Since the representation (n,m)∈N b nm (e n ⊗ e m ) of the tensor w converges with respect to the square ordering, the coordinates b nm form a zero sequence. Therefore, for each given δ > , we find an N ∈ N such that b nn < δ whenever n ≥ N . Subtracting the vector ( , δ (e N ⊗e N )) from ( , w), we obtain an arbitrarily near vector to ( , w) that is not contained in the convex hull, since one coordinate on the diagonal is strictly negative. Thus, the convex hull of ( ) has an empty interior.
Proposition . (Continuity of the representative). LetF : X × (X ⊗ π,sym X ) → R be a representative of the diconvex mapping F : X → R on the real finite-dimensional B space X . If F is finite on a non-empty, open set, then the representativeF is continuous in the interior of the effective domain dom(F ).
Proof. By assumption, there is a point u ∈ X such that F is finite on an ϵ-ball B ϵ (u) around u for some ϵ > . Consequently, the representativeF is finite on the set ( ) Using the construction in the proof of Lemma . , we find a simplex with vertices in ( ) that contains an inner point ( , w) of the convex hull of ( ) and hence of the effective domain dom(F ). SinceF is convex and finite on the vertices of the constructed simplex, the representativeF is bounded from above on a non-empty, open neighbourhood around ( , w), which is equivalent to the continuity ofF on the interior of the effective domain dom(F ), see for instance [ET , Sho ].
On the basis of this observation, we obtain the following computation rules for the representative subdifferential on finite-dimensional B spaces, which follow im-mediately form Proposition . and the classical sum and chain rules, see for instance [BC , ET , Roc , Sho ].
Proposition . (Representative sum rule). Let F : X → R and G : X → R be diconvex functionals on the real finite-dimensional B space X with representativesF andG. If there exists a non-empty, open set where F and G are finite, then for all u in X with respect to the representativeF +G of F + G.
Proof. In the proof of Proposition . , we have constructed an appropriate simplex to prove the existence of a point ( , w) in X × (X ⊗ π,sym X ) where the representativeF is finite and continuous. Using the same simplex again, we see that the representativeG is finite and continuous in the same point ( , w). Applying the classical sum rule -see for instance [Sho , Proposition II. . ] -to the representativeF +G, we obtain for all (u, u ⊗ u) in X × (X ⊗ π,sym X ) and thus the assertion.
Remark . . In order to apply the classical sum rule in the proof of Proposition . , it would be sufficient if only one of the functionalsF andG is continuous in ( , w). The assumption that F and G are finite at some non-empty, open set is thus stronger than absolutely necessary. On the other side, this assumption is needed to ensure that the effective domain ofG and the interior of the effective domain ofF have some point in common.
Proposition . (Representative chain rule for linear operators). Let K : X → Y be a bounded, surjective linear mapping and F : Y → R be a diconvex functional with representativeF on the real B spaces X and Y . If Y is finite-dimensional, and if there exists a non-empty, open set where F is finite, then Proof. Like in the proof of Proposition . , the mappingF •(K ×(K ⊗ π,sym K)) is a proper, convex representative of F • K. Since the linear operator K is surjective, the mapping K ⊗ π,sym K is surjective too. In more detail, there exists finitely many vectors e , . . . , e N such that the images f n = K(e n ) form a basis of the finite-dimensional B space Y . Since the symmetric tensors , Proposition . ], the bounded linear mapping K ⊗ π,sym K is also surjective. Using Proposition . , we thus always find a point (K × (K ⊗ π,sym K))( , w) whereF is continuous. Now, the classical chain rule -see for instance [Sho , Proposition II. . ] -implies for all u in X , which establishes the assertion.
Remark . . In the proof of Proposition . , the surjectivity of K implies the nonemptiness of the intersection between the interior of dom(F ) and the range of K × (K ⊗ π,sym K). So long as this intersection is not empty, Proposition . remains valid even for non-surjective operators K. The non-emptiness of the intersection then depends on the representativeF and the mapping K × (K ⊗ π,sym K). The intention behind Proposition . has been to give a chain rule that only depends on properties of the given F and K.
Proposition . (Representative chain rule for convex functionals). Let K : X → Y be a bounded dilinear mapping and F : Y → R be a proper, convex functional on the real B spaces X and Y . If there exists a non-empty, open set where F is bounded from above, and if the interior of the effective domain dom(F ) and the range ran(K) are not disjoint, then∂ Proof. Since the proper and convex mapping F is bounded from above on some nonempty, open set, the function F is continuous on the interior its effective domain dom F , see for instance [ET , Proposition . ]. Consequently, we always find a pointK( , ⊗ ) where F is continuous, which allows us to apply the classical chain rule -see for instance [Sho , Proposition II. . ] -to the representative F •K. In this manner, we obtain for all u in X .
Although the dilinear and the representative subdifferential of a diconvex functional F coincide with respect to the representative conv F ⊗ , the established computation rules for the representative subdifferential cannot be transferred to the dilinear subdifferential in general. A counterexample for the sum rule is given below. The main reason for this shortcoming is that the convexification of (F + G) ⊗ does not have to be the sum of the convexifications of F ⊗ and G ⊗ . An analogous problem occurs for the composition.
Counterexample . . One of the simplest counterexamples, where the sum rule is failing for the dilinear subdifferential, is the sum of the absolute value function | · | : R → R and the indicator function χ : R → R of the interval [− , ] given by +∞ else.
As mentioned above, for a function from R into R, the dilinear subdifferential consists of all parabolae beneath that function at a certain point. Looking at the point zero, we have the dilinear subdifferentials The sum of the dilinear subdifferentials thus consists of all parabolae with leading coefficient c ≤ and linear coefficient c ∈ [− , ]. However, the dilinear subdifferential of the sum also contains parabolae with positive leading coefficient, see the schematic illustrations in Figure . Generalizing the classical F rule, see for instance [BC ], we obtain a necessary and sufficient optimality criterion for the minimizer of a diconvex functional based on the dilinear subdifferential calculus.
Proof. By definition, zero is contained in the dilinear subgradient of F at u * if and only if F ( ) ≥ F (u * ) for all in X , which is equivalent to u * being a minimizer of F .
Using the dilinear subgradient, we now generalize the classical B distance, see for instance [BC , IJ ], to the dilinear/diconvex setting.
distance of and u with respect to F and (ξ , Ξ) is given by

. Dilinear inverse problems
During the last decades, the theory of inverse problems has become one of the central mathematical tools for data recovery problems in medicine, engineering, and physics. Many inverse problems are ill posed such that finding numerical solutions is challenging. Although the regularization theory of linear inverse problems is well established, especially with respect to convergence and corresponding rates, solving non-linear inverse problems remains problematic, and many approaches depend on assumptions and source conditions that are difficult to verify or to validate. Based on the tensorial lifting, we will show that the linear regularization theory on B spaces can be extended to the non-linear class of dilinear inverse problems.
To be more precise, we consider the T regularization for the dilinear inverse problem where K : X → Y is a bounded dilinear operator between the real B spaces X and Y , and where † denotes the given data without noise. In order to solve this type of inverse problems, we study the T functional where δ represent a noisy version of the exact data † , where p ≥ , and where R is some appropriate diconvex regularization term. To verify the well-posedness and the regularization properties of the T functional α , we rely on the well-established non-linear theory, see for instance [HKPS ]. For this, we henceforth make the following assumptions, which are based on the usual requirements for the linear case, cf. [IJ , Assumption . ].
Assumption . . Let X and Y be real B spaces with predual X * and Y * , where X * is separable or reflexive. Assume that the data fidelity functional S(·) ≔ || K(·) − δ || p with the dilinear mapping K : X → Y and the non-negative, proper, diconvex regularization functional R : X → R satisfy: functional α is coercive in the sense that α (u) → +∞ whenever ||u || → +∞.
(ii) The functional R is sequentially weakly * lower semi-continuous.
(iii) The dilinear operator K is sequentially weakly * continuous.
Remark . . Since X * is a separable or reflexive B space, we can henceforth conclude that every bounded sequence (u n ) n ∈N in X contains a weakly * convergent subsequence, see for instance [Meg , Theorem . . ] and [Lax , Theorem . ] respectively.
For the non-linear regularization theory in [HKPS ], which covers a much more general setting, the needed requirements are much more sophisticated and comprehensive. Therefore, we briefly verify that Assumption . is compatible with these requirements.
Lemma . (Verification of required assumptions). The requirements on the dilinear operator K and the regularization functional R made in Assumption . satisfy the requirements in [HKPS , Assumption . ].
Proof. We verify the six required assumptions in [HKPS ] step by step.
(i) We have to equip the B spaces X and Y with topologies τ X and τ Y weaker than the norm topology. Since X and Y have predual spaces, we simply associate the related weak * topologies.
(ii) The norm of Y has to be sequentially lower semi-continuous with respect to τ Y , which immediately follows from the weak * lower semi-continuity of the norm, see for instance [Meg , Theorem . . ].
(iii) The forward operator has to be sequentially continuous with respect to the topologies τ X and τ Y , which coincides with Assumption . .iii.
(iv) The regularization functional R has to be proper and sequentially lower semicontinuous with respect to τ X , which coincides with Assumption . .ii.
(v) The domain of the forward operator is sequentially closed with respect to τ X , and the intersection dom(K) ∩ dom(R) is non-empty, where dom(K) denotes the domain of definition of K. Both assumptions are satisfied since K is defined on the entire B space X , which is sequentially weakly * complete, see for instance [Meg , Corollary . . ], and since R is a proper functional.
(vi) For every α > and M > , the sublevel sets have to be sequentially compact with respect to τ X . By Assumption . .i, the T functional α is coercive, which implies that the sublevel sets M α (M) are bounded for every α > and M > . Since the functional α is weakly * lower semi-continuous, the sublevel sets M α (M) are also sequentially weakly * closed. Due to the assumption that X * is separable or reflexive, the sequential weak * compactness follows, see Remark . .
Remark . . Since the original proofs of the well-posedness, stability, and consistency only employs the sequential versions of the weak * (semi-)continuity, closure, and compactness, we have weakened these assumptions accordingly. Likewise, we have skipped the unused convexity of the regularization functional.
Since the dilinear operator K is the composition of u → (u, u ⊗ u) and the lifted operatorK, the weak * continuity in Assumption . .iii may be transferred to the rep-resentativeK. The main problem with this approach is that the predual of the tensor product X ⊗ π,sym X does not have to exist, even if the predual X * of the real B space X is known. Therefore, we equip the symmetric tensor product X ⊗ π,sym X with an appropriate topology.
As discussed above, each finite-rank tensor in ω ∈ X * ⊗ X * defines a bilinear form B ω : X × X → R. If we consider the closure of X * ⊗ X * with respect to the norm , we obtain the injective tensor product X * ⊗ ϵ X * , see [DF , Rya ]. Since the space of bilinear forms B(X × X ) is the topological dual of X ⊗ π X , the injective tensor product X * ⊗ ϵ X * of the predual X * is a family of linear functionals on X ⊗ π X . If the B space X has the approximation property, i.e. the identity operator can be approximated by finite-rank operators on compact sets, then the canonical mapping from X ⊗ π X into (X * ⊗ ϵ X * ) * becomes an isometric embedding, see [Rya , Theorem . ]. In this case, the injective tensor product X * ⊗ ϵ X * even defines a separating family, and the projective tensor product X ⊗ π X together with the topology induced by the injective tensor product X * ⊗ ϵ X * thus becomes a H space. In our setting, the separation property and the approximation property are even equivalent, cf. In the same manner, the subspace X * ⊗ ϵ,sym X * of the symmetric tensors ω -of the symmetric bilinear forms B ω -forms a family of linear functionals on X ⊗ π,sym X . If the B space X has the approximation property, the symmetric injective tensor product again defines a separating family.
On the basis of this observation, we equip the symmetric projective tensor product X ⊗ π,sym X with the weakest topology such that each tensor ω of the symmetric injective tensor product X * ⊗ ϵ,sym X * becomes a continuous, linear functional. More precisely, the weak * topology induced by X * ⊗ ϵ,sym X * is generated by the family of preimages [Meg , Proposition . . ]. Since the symmetric injective tensor product X * ⊗ ϵ,sym X * is a subspace of all linear functionals on X ⊗ π,sym X , the induced topology is locally convex, see [Meg , Theorem . . ]. Further, a sequence (w n ) n ∈N of tensors in X ⊗ π,sym X converges to an element w in X ⊗ π,sym X with respect to the topology induced by X * ⊗ ϵ,sym X * weakly * if and only if ω(w n ) converges to ω(w) for each ω in X * ⊗ ϵ,sym X * , see for instance [Meg , Proposition . . ].
The central reason to choose the injective tensor product X * ⊗ ϵ,sym X * as topologizing family for X ⊗ π,sym X is that, under further assumptions, like the H space setting X = H , the injective tensor product actually becomes a true predual of the projective tensor product, see [Rya ].
Lemma . (Weak * continuity of the tensor mapping). Let X be a real B space with predual X * . The mapping ⊗ : X → X ⊗ π,sym X with u → u ⊗ u is sequentially weakly * continuous with respect to the topology induced by the injective tensor product X * ⊗ ϵ,sym X * .
Proof. Let ϕ ⊗ ϕ be a rank-one tensor in X * ⊗ ϵ,sym X * , and let (u n ) n ∈N be a weakly * convergent sequence in the B space X . Without loss of generality, we postulate that the sequence (u n ) n ∈N is bounded by ||u n || ≤ . Under the assumption that u is the weak * limit of (u n ) n ∈N , we observe Obviously, this observation remains valid for all finite-rank tensors in X * ⊗ ϵ,sym X * . Now, let ω be an arbitrary tensor in X * ⊗ ϵ,sym X * . For every ϵ > , we find a finite-rank approximation ω of the tensor ω such that || ω − ω || ϵ ≤ ϵ / . Hence, for suitable large n, for all linear functionals ω in X * ⊗ ϵ,sym X * .
Proposition . (Sequential weak * continuity of a dilinear operator). Let X and Y be real B spaces with preduals X * and Y * . If the representativeK : X × (X ⊗ π,sym X ) → Y is sequentially weakly * continuous with respect to the weak * topology on X and the weak * topology on X ⊗ π,sym X induced by X * ⊗ ϵ,sym X * , then the related dilinear operator K : X → Y is sequentially weakly * continuous.
Proof. Since the related dilinear operator K is given by K(u) ≔K(u, u ⊗u), the assertion immediately follows from the sequential weak * continuity ofK and the sequential weak * continuity of the tensor mapping u → (u, u ⊗ u), cf. Lemma . .
With an entirely analogous argumentation, the required sequential weak * lower semicontinuity of the regularization functional R may be inherit from the sequential weak * lower semi-continuity of the representativeȒ.
Proposition . (Sequential weak * lower semi-continuity of a diconvex mapping). Let X be a real B space with predual X * . If the representativeF : X × (X ⊗ π,sym X ) → R is sequentially weakly * lower semi-continuous with respect to the weak * topology on X and the weak * topology on X ⊗ π,sym X induced by X * ⊗ ϵ,sym X * , then the related diconvex mapping F : X → R is sequentially weakly * lower semi-continuous.
Proof. Being a composition of the sequentially weakly * continuous tensor mapping u → (u, u ⊗ u), cf. Lemma . , and the sequentially weakly * lower semi-continuous representativeF , the related diconvex mapping F given by F (u) ≔F (u, u⊗u) is obviously sequentially weakly * continuous.
At this point, one may ask oneself whether each sequentially weakly * lower semicontinuous, diconvex mapping possesses a sequentially weakly * lower semi-continuous, convex representative. At least for finite-dimensional spaces, where the weak * convergence coincides with the strong convergence, this is always the case. Remembering that all real d-dimensional B spaces are isometrically isomorphic to R d , we can restrict our argumentation to X = R d equipped with the E ian inner product and norm. Further, the sequential weak * lower semi-continuity here coincides with the lower semicontinuity. The projective tensor product R d ⊗ π,sym R d becomes the space of symmetric matrices R d×d sym equipped with the H -S inner product and the F norm. Moreover, the space R d×d sym is spanned by the rank-one tensors u ⊗ u = u u T with u ∈ R d . The dual space (R d ⊗ π,sym R d ) * may also be identified with the space of symmetric matrices R d×d sym .

Theorem . (Lower semi-continuity in finite dimensions). A diconvex mapping F : R d → R is lower semi-continuous if and only if there exists a lower semi-continuous
Proof. In Proposition . , we have in particular shown that the lower semi-continuity of F implies the lower semi-continuity of F . Thus, it only remains to prove that each lower semi-continuous diconvex functional possesses a lower semi-continuous representativȇ F . The assertion is obviously true for the constant functional F ≡ +∞ with representativȇ F (u, w) ≔ for one point (u, w) with w u ⊗ u andF (·, ·) = +∞ otherwise. For the remaining functionals, the central idea of the proof is to show that the lower semicontinuous convexification conv F ⊗ -the closure of the convex hull conv F ⊗ -of the mapping F ⊗ in ( ) is a valid representative of a lower semi-continuous mapping F , which means that conv F ⊗ (u, u ⊗u) = F (u) for all u ∈ R d . For the sake of simplicity, we assume that F (u) ≥ and thus F ⊗ (u, u ⊗ u) ≥ for all u in R d , which can always be achieved by subtracting a (continuous) dilinear minorant, see Remark . .
For a fixed point (u, u ⊗ u) on the diagonal, we distinguish the following three cases: (i) The point (u, u ⊗u) is not contained in the relative closure of dom(conv F ⊗ ), which implies F (u) = conv F ⊗ (u, u ⊗ u) = +∞.
(ii) The point (u, u ⊗u) lies in the relative interior of dom(conv F ⊗ ). Since the effective domain dom(conv F ⊗ ) is a subset of conv{(u, u ⊗ u) : u ∈ R d }, and since (u, u ⊗ u) is an extreme point of the latter set, see Lemma . , the point (u, u ⊗ u) thus has to be extreme with respect to the effective domain. Since the extreme points of a convex set are, however, contained in the relative boundary except for the zerodimensional case, see [Roc , Corollary . . ], the effective domains of conv F ⊗ and F ⊗ have to consist exactly of the considered point (u, u ⊗ u). In this instance, the closed convex hull conv F ⊗ equals F ⊗ and is trivially a sequentially weakly * continuous representative.
(iii) The point (u, u ⊗ u) is contained in the relative boundary of dom(conv F ⊗ ).
To finish the proof, we have to show F (u) = conv F ⊗ (u, u ⊗ u) for the third case.
In order to compute conv F ⊗ (u, u ⊗ u), we apply [Roc , Theorem . ], which implies where ( , w) is some point in the non-empty relative interior of dom(conv F ⊗ ). Next, we take a sequence ( k , w k ) ≔ λ k (u, u ⊗ u) + ( − λ k )( , w) with λ k ∈ ( , ) so that lim k→∞ λ k = and consider the limit of the function values conv F ⊗ ( k , w k ). Since the complete sequence ( k , w k ) ∞ k= is contained in the relative interior of dom(conv F ⊗ ), cf. [Roc , Theorem . ], all functions values conv F ⊗ ( k , w k ) are finite and can be approximated by C 's theorem. More precisely, for every ρ > and for every k ∈ N, we can always find convex combinations ( k , n is in [ , ] so that N + n= α (k) n = , and where N is an integer not greater than the dimension of R d × R d×d sym , such that In the next step, we examine the occurring sequence of convex combinations in more detail. For this purpose, we define the half spaces where I denotes the identity matrix. Obviously, a vector ( , ⊗ ) is contained in the Similarly, the vector ( , ⊗ ) is contained in H − u,ϵ +(u, u ⊗u) if and only if || − u || ≤ ϵ. As mentioned above, we now consider a sequence of convex combinations where α (k) n ∈ [ , ] so that N + n= α (k) n = , and where N is some fixed integer independent of k. For n = , either the sequence (u (k) ) k ∈N has a subsequence converging to u or there exists an ϵ > such that ||u (k) − u || ≥ ϵ for all k in N. In the second case, the complete sequence (u (k) ) k ∈N is contained in the shifted half space H + u,ϵ + (u, u ⊗ u). Thinning out the sequence of convex combinations by repeating this construction for the remaining indices n = , . . . , N iteratively, we obtain a subsequence of the form where we assume that the first L sequences (u (ℓ) n ) ℓ ∈N possess the accumulation point u without loss of generality -if necessary, we rearrange the indices n = , . . . , N + accordingly. Thinning out the subsequence even further, we can also ensure that the coefficients (α (ℓ) n ) ℓ ∈N converge for every index n. In the following, we have to take special attention of the subsequence (u (ℓ) n ) ℓ ∈N not converging to u. Therefore, we consider the case where β (ℓ) ≔ N + n=L+ α (ℓ) n does not become constantly zero after some index ℓ more precisely. Taking a subsequence with β (ℓ) , and re-weighting the sequence of convex combinations, we now obtain where ϵ > is chosen smaller than ϵ L+ , . . . , ϵ N + . Since the second sum is a convex combination of points in H + u,ϵ + (u, u ⊗ u), the value of the sum is also contained in is not contained in the closed set H + u,ϵ + (u, u ⊗ u) for ϵ > by construction, the coefficients β (ℓ) could neither become constantly one. For an appropriate subsequence, the re-weighted convex combinations are thus well defined. Obviously, the first sum converges to (u, u ⊗ u). If we now assume that the sequence (β (ℓ) ) ℓ ∈N does not converge to zero, the second sum has to converge to some point in H + u,ϵ + (u, u ⊗ u). Consequently, (u, u ⊗ u) is a non-trivial convex combination of itself with a further point in H + u,ϵ +(u, u ⊗u). Since (u, u ⊗u) is not contained in H + u,ϵ +(u, u ⊗u), see ( ), this is not possible. Hence, the coefficient β (ℓ) converges to zero, which is our main observation in this step.
Applying the subsequence construction above to the sequence of function values and exploiting the lower semi-continuity and non-negativity of F , we can finally estimate the limit of the function values conv F ⊗ ( k , w k ) by because − β (ℓ) = L n= α (ℓ) n converges to one as discussed above. Since the accuracy ρ of the approximation can be chosen arbitrarily small, we thus have Hence, convF ⊗ (u, u ⊗ u) equals F (u) for all u ∈ R d , and the lower semi-continuous convex hull conv F ⊗ is a valid representative of F .
Remark . . Using an analogous argumentation, one may extend Theorem . to the lower semi-continuous convex hull conv F ⊗ . More precisely, if F : R d → R is a lower semi-continuous, diconvex mapping, then the representative subdifferential with respect to conv F ⊗ and the dilinear subdifferential coincide, i.e.

. . Well-posedness and regularization properties
We now return to the well-posedness, stability, and consistency of the T regularization of the dilinear inverse problem K(u) = † . In other words, we study the regularization properties of the variational regularization Since the introduced dilinear regularization is particularly an instance of the non-linear T regularization in B spaces, see for instance [HKPS ], the well-posedness, stability, and consistency immediately follow from the well-established non-linear regularization theory. For the sake of completeness and convenience, we briefly summarize the central results with respect to our setting.
Firstly, the T functional α in ( ) is well posed in the sense that the minimum of the regularized problem min{ α (u) : u ∈ X } is attained, and that the related minimizer is thus well defined.
Proof. The well-posedness immediately follows from [HKPS , Theorem . ] since the required assumptions are fulfilled by Lemma . .
Stability of a variational regularization method means that the minimizers u δ α of the T functional α weakly * depend on the noisy data δ . If the regularization functional R satisfies the so-called H-property, see for instance [IJ , Wer ], the dependence between the solution of the regularized problem and the corrupted data is actually strong.
Definition . (H-property). A functional R : X → R possesses the H-property on the space X if any weakly * convergent sequence (u n ) n ∈N in X with limit u and with R(u n ) → R(u) converges to u strongly.
Theorem . (Stability). Let the sequence ( n ) n ∈N in Y be convergent with limit δ ∈ Y . Under Assumption . , the sequence (u n ) n ∈N of minimizers of the T functional α in ( ) with n in place of δ contains a weakly * convergent subsequence to a minimizer u δ α of α . If the minimizer of α is unique, then the complete sequence (u n ) n ∈N converges weakly * . If the functional R possesses the H-property, then the sequence (u n ) n ∈N converges in norm topology.
Proof. The existence of a weakly * convergent subsequence, whose limit is a minimizer of α , follows directly from the stability of the non-linear T regularization [HKPS , Theorem . ]. Again, the required assumptions are fulfilled by Lemma . . Moreover, if the minimizer u δ α is unique, each subsequence of (u n ) n ∈N contains a subsequence weakly * converging to u δ α , and hence, the entire sequence (u n ) n ∈N converges weakly * , cf. [IJ , Theorem . ].
Completely analogously to the proof of [IJ , Theorem . ], one can now show that the sequence R(u n ) must converge to R(u * ). In this case, the H-property additionally yields the convergence of (u n ) n ∈N in norm topology.
Finally, the T regularization of a dilinear inverse problem is consistent; so the minimizer u δ α weakly * converges to a solution u † of the unperturbed problem K(u) = † if the noise level δ goes to zero. More precisely, the solution u δ α converges to an Rminimizing solution, see for instance [HKPS , Definition . ] or [IJ , Definition . ] for the definition.

Definition . (R-minimizing solution).
for all further solutions u of the dilinear equation K(u) = † .
Theorem . (Existence of an R-minimizing solution). Under Assumption . , there exists at least one R-minimizing solution to the dilinear problem K(u) = † .
Proof. The existence of an R-minimizing solution immediately follows from [HKPS , Theorem . ]. The needed assumptions are satisfied by Lemma . .
If the R-minimizing solution u † is unique, then the entire sequence (u δ n α n ) n ∈N converges weakly * . If the functional R possesses the H-property, then the sequence (u δ n α n ) n ∈N converges in norm topology.
Proof. Due to the consistency property of the non-linear T regularization, see [HKPS , Theorem . ], the sequence of minimizers (u δ n α n ) n ∈N possesses a subsequence converging to an R-minimizing solution weakly * . Once more, the required assumption in [HKPS ] are implied by Lemma . .
If the R-minimizing solution u † is unique, each subsequence of (u δ n α n ) n ∈N has a weakly * converging subsequence. Consequently, the entire sequence (u δ n α n ) n ∈N converges weakly * . As verified in the proof of [HKPS , Theorem . ], the sequence R(u n ) converges to R(u * ). Under the assumption that the regularization functional R has the H-property, the convergence of (u δ n α n ) n ∈N is thus strong.

. . Convergence analysis under source-wise representations
Based on the dilinear subdifferential calculus, we now analyse the convergence behaviour of the variational regularization method for dilinear inverse problems K(u) = † if the noise level δ goes to zero. In the following, we assume that the dilinear operator K : X → Y maps from a real B space into a real H space Y . Further, we restrict ourselves to the squared H norm || · || as data fidelity functional S in ( ), which means that we consider the T functional Looking back at Theorem . , we recall that the dilinear inverse problem K(u) = † always possesses an R-minimizing solution u † with respect to the regularization functional R. Consequently, F 's rule (Theorem . ) implies that zero must be contained in the dilinear subdifferential ∂ β (R + χ {K (u)= † } )(u † ), where the indicator function χ {K (u)= † } (u) is if u is a solution of the inverse problem K(u) = † and +∞ else. Applying the sum and chain rule in Proposition . and . , we have Without further assumptions, the sum and chain rule here only yield an inclusion. Although we always have ∈ ∂ β (R + χ {K (u)= † } )(u † ), we hence cannot guarantee ∈ ranK * + ∂ β R(u † ) or, equivalently, ranK * ∩ ∂ β R(u † ) ∅. Against this background, we postulate the regularity assumption that the range of the adjoint operator and the dilinear subdifferential are not disjoint. In other words, we assume the existence of a source-wise representationK Theorem . (Convergence rate). Let u † be an R-minimizing solution of the dilinear inverse problem K(u) = † . Under the source conditionK * ω = (ξ † , Ξ † ) ∈ ∂ β R(u † ) for some ω in Y and under Assumption . , the minimizers u δ α of the T regularization α in ( ) converges to u † in the sense that the dilinear B distance between u δ α and u † with respect to the regularization functional R is bounded by and the data fidelity term by Proof. Inspired by the proof for the usual subdifferential in [IJ ], the desired convergence rate for the dilinear subdifferential can be established in the following manner. Since u δ α is a minimizer of the T functional α in ( ), u δ α and u † satisfy Remembering K(u † ) = † , we can bound the norm on the right-hand side by || † − δ || ≤ δ . Rearranging the last inequality and exploiting the source condition, we get Rearranging the terms, completing the square, and applying C -S 's inequality, we obtain which proves the convergence rate for the B distance. The second convergence rate follows immediately by applying the reverse triangle inequality.

. Deautoconvolution problem
In order to give a non-trivial example for the practical relevance of the developed dilinear regularization theory, we consider the deautoconvolution problem, where one wishes to recover an unknown signal u : R → C with compact support from its kernel-based autoconvolution where k : R → C is an appropriate kernel function. Problems of this kind occur in spectroscopy, optics, and stochastics for instance, see [BH ].
Following the model in [BH ], we assume that u is a square-integrable complexvalued signal on the interval [ , ], or, in other words, u ∈ L C ([ , ]). To ensure that the integral in ( ) is well defined, we extend the signal u outside the interval [ , ] with zero and restrict ourselves to bounded kernels k . Considering the support of u, we may moreover assume supp k ⊂ (s, t) : ≤ s ≤ and s ≤ t ≤ s + ( ) and k(s, t) = k(t − s, t) ( ≤ s ≤ , s ≤ t ≤ s + ).
( ) The symmetry property ( ) can be demanded in general because of the identity After these preliminary considerations, we next verify that the kernel-based autoconvolution A k is a bounded dilinear operator. For this, we exploit that the autoconvolution A k is the quadratic mapping related to the symmetric bilinear mapping The well-definition and boundedness of B k immediately follows from J 's inequal-ity. More precisely, we obtain In connection with Example . , this shows that the kernel-based autoconvolution A k is a bounded dilinear mapping from L C ([ , ]) into L C ([ , ]). In order to apply the developed dilinear regularization theory, we interpret the complex H spaces L C ([ , ]) and L C ([ , ]) as real H spaces with the inner product ·, · R ≔ ℜ ·, · C . Although the deautoconvolution problem is ill posed in general, the unperturbed in- Proof. In difference to [GHB + ], our proof is mainly based on F analysis. Using the (F ) convolution theorem, we notice that u is a solution of the deautoconvo- is constantly zero, the assertion is trivially true. Otherwise, the F transforms of the compactly supported signals u and † are restrictions of entire functions by the theorem of P -W , and the related entire functions are completely determined by F[u] and F[ † ]. Therefore, we may always find a point ω such that F[u] and F[ † ] are non-zero in an appropriately small neighbourhood U around ω . On this neighbourhood, there exist exactly two roots of F[ † ], which are restrictions of holomorphic functions, see for instance [FL , Section ]. Consequently, the Fourier transform of u is either for all ω ∈ U . Extending F[u] from the neighbourhood to the whole real line by using the unique, corresponding entire function, one can conclude that u and −u are the only possible solutions.
We proceed by considering the perturbed deautoconvolution problem Since the autoconvolution is a bounded dilinear mapping, we can apply the developed dilinear regularization theory to the T functional where we choose the squared norm of L C ([ , ]) as regularization term. In order to verify that the autoconvolution fulfils the required assumptions and to analyse the sourcewise representation ( ), firstly, we need a suitable representation of the kernel-based autoconvolution and of the dual and predual spaces of L C ([ , ]) ⊗ π,sym L C ([ , ]).
For this, we notice that each tensor w in the tensor product H ⊗ π H , where H is an arbitrary H space, may be interpreted as a nuclear operator in sense of the mapping L w or R w in ( -). 'Nuclear' here means that the singular values of L w and R w are absolutely summable. More precisely, the projective tensor product of a H space with itself is isometrically isomorphic to the space of nuclear operators from H to H , i.e.
H ⊗ π H ≃ N(H ), see for instance [Wer , Section VI. ]. Further, the injective tensor product is here isometrically isomorphic to the space of compact operators, i.e.
see [Rya , Corollary . ], and thus becomes a true predual of the projective tensor product, which means or by the H -S inner product if w is interpreted as nuclear operator. The corresponding spaces for the symmetric projective tensor product H ⊗ π,sym H coincide with the related self-adjoint operators. In particular, these identifications are valid for the real H space L C ([ , ]).
To handle the autoconvolution A k , we split this mapping into a linear integral part More precisely, we employ the factorization A k = I k • ⊙ where the operators I k and ⊙ are given by On the basis of these mappings, we can determine the lifting of the autoconvolution A k .
Lemma . (Li ing of the autoconvolution). The unique (quadratic) lifting of the kernel-based autoconvolution A k = I k • ⊙ is given by For the dilinear lifting, we here use a matrix-vector-like representation. More precisely, the mapping ( ,Ȃ k ) is defined by with u ∈ L C ([ , ]) and w ∈ L C ([ , ]) ⊗ π,sym L C ([ , ]).
Proof of Lemma . . Firstly, we notice that the related bilinear mapping of the quadratic mapping ⊙ is given by for arbitraryu and in L C ([ , ]). Obviously, this bilinear map is bounded with norm one, which means that there exists a unique bilinear lifting, see Proposition . . Restricting this lifting to the symmetric projective tensor product, we obtain the unique quadratic lifting⊙ : Since the integral operator I k is bounded with norm ||k || ∞ , the complete mappingȂ k = I k •⊙ is bounded too. Moreover, we havȇ so the defined mappingȂ k is the unique quadratic lifting of the autoconvolution A k . In Example . , we have already seen that the unique dilinear lifting with domain of definition L C ([ , ]) × (L C ([ , ]) ⊗ π,sym L C ([ , ])) of an arbitrary quadratic mapping is completely independent of the first component space. In other words, the first component is always mapped to zero, which completes the proof.
Remark . . Although the mapping⊙ looks like a continuous embedding of the symmetric projective tensor product L C ([ , ]) ⊗ π,sym L C ([ , ]) into L ([ , ] ), this is not the case. Since we consider L C ([ , ]) as an R-linear space, the functions u and iu are orthogonal and thus linear independent. The reason for this is simply that i is not a real scalar. As a consequence, for u , the tensors u ⊗ u and iu ⊗ iu are also linear independent, see [Rya , Proposition . ]. Now, the image of u ⊗ u + iu ⊗ iu is the zero function, which shows that⊙ is not injective. If one restricts oneself to the function space of real functions L R ([ , ]), or if one consider L C ([ , ]) as C-linear space, then⊙ truly becomes a continuous embedding.
With the factorization of the kernel-based autoconvolution in mind, we next determine the related adjoint operator of the liftingȂ k .

Lemma . (Adjoint of the autoconvolution). The adjoint operatorȂ
where I * k is a mapping between L C ([ , ]) and L C, sym ([ , ] ), and the adjoint operator of the dilinear lifting by For the adjoint dilinear lifting, we again use a matrix-vector-like representation. In more detail, the mapping ( ,Ȃ * k ) T is defined by Proof of Lemma . . The representation of the adjoint integral operator I * k can be directly verified by ). The remaining assertion immediately follows from the properties of the adjoint.
Remark . . The lifted operator⊙ maps every tensor in the symmetric projective tensor product to a square-integrable function. The adjoint mapping⊙ * thus allows us to interpret each function ω in L C, sym ([ , ] ) as a bounded linear functional on the projective tensor product L C ([ , ]) ⊗ π,sym L C ([ , ]). More precisely, the action of the function ω on an arbitrary tensor w is given by which we will exploit later.
Remark . . For the trivial kernel k ≡ , the image of the adjoint operator I * k consists of all functions w in L C, sym ([ , ] ) with H structure, where the diagonal t → w(t, t) coincides with the given ϕ up to scaling.
The next ingredients for the dilinear regularization theory in Section are the required conditions in Assumption . . As a preparing step, we prove that the lifted quadratic operator⊙ is sequentially weakly * continuous.
Proof. The central idea of the proof is to exploit Proposition . , which means that we have to show the sequential weak * continuity of the dilinear lifting⊙ with respect to the topology induced by L C ([ , ]) ⊗ ϵ,sym L C ([ , ]). Since the dilinear lifting of a quadratic operator is independent of the first component space, cf. Example . , it is enough to show the sequential weak * continuity of the quadratic lifting⊙ in Lemma . . For this, we have to show ω,⊙[w n ] R → ω,⊙[w] R ( ) for every ω in L C, sym ([ , ] ), and for every weakly * convergent sequence w n * − ⇀ w. As mentioned above, the symmetric injective tensor product is here isometrically isomorphic to the self-adjoint compact operators, where the action of a self-adjoint compact operator Φ is given by the lifting of the quadratic form (u, u) → Φ[u], u R . This observation is the key component to establish the assertion. More precisely, if we can show that the action of an arbitrary symmetric function ω see Remark . , which is equivalent to the testing in ( ), corresponds to the lifting of a self-adjoint compact operator, then the assertion is trivially true.
For this purpose, we consider the action of ω to a symmetric rank-one tensor u ⊗ u, which may be written as The central observation is that the R-linear operator Φ ω resembles a F integral operator. Due to the occurring conjugation, for a fixed point (s, t), the multiplication with w(s, t) here only acts as an R-linear mapping instead of an usually C-linear mapping on u(s). Similarly to the classical theory, see for instance [Wer ], we can approximate the kernel function ω by a sequence of appropriate step functions ω n on a rectangular partition of [ , ] such that ω n → ω in L C, sym ([ , ] ). The ranges of the related operators Φ ω n are finite-dimensional, and because of the convergence of Φ ω n to Φ ω , the compactness of Φ ω follows. Since ω is symmetric, the operator Φ ω is moreover self-adjoint. Consequently, the quadratic lifting w → ω, For a weakly * convergent sequence w n * − ⇀ w, we thus have ( ) for all ω in L C ([ , ] ), which shows the sequential weak * continuity of the quadratic lifting⊙ and hence of the dilinear lifting ( ,⊙). The sequential weak * continuity of the dilinear operator ⊙ now immediately follows from Proposition . .
Using Lemma . , we can now verify that the T regularization of the deautoconvolution problem satisfies the required assumptions for the developed regularization theory in Section .
Lemma . (Verification of required assumptions). The T functional α in ( ) related to the kernel-based autoconvolution A k fulfils the requirements in Assumption . .

Proof.
We briefly verify the needed requirements step by step.
(i) Obviously, the T functional α in ( ) is coercive since the regularization term coincides with the squared H space norm.
(ii) For the same reason, the regularization term is sequentially weakly * lower semicontinuous, see for instance [Meg , Theorem . . ].
(iii) In Lemma . , we have already proven that the quadratic operator ⊙ of the factorization A k = I k • ⊙ is sequentially weakly * continuous. Further, the obvious norm-to-norm continuity of the integral operator I k implies the weak-to-weak continuity of I k , see for instance [Meg , Proposition . . ]. Since L C ([ , ] ) and L C ([ , ]) are H spaces, the mapping I k is weakly * -to-weakly * continuous as well. Consequently, the composition A k = I k • ⊙ is sequentially weakly * continuous as required.
Remark . . Since the T functional α for the deautoconvolution problem fulfils all constraints in Assumption . by Lemma . , we can employ the developed dilinear/diconvex regularization theory in Section . Therefore, the related minimization problem minimize is well posed, stable, and consistent.
Besides the well-known well-posedness, stability, and consistency of the regularized deautoconvolution problem, the convergence rate introduced by Theorem . is far more interesting. In order to study the employed source condition, we have to compare the range of the adjoint autoconvolution operator ( ,Ȃ * k ) T , see Lemma . , and the dilinear subdifferential of the squared H norm on L C ([ , ]). More generally, we initially determine the dilinear subdifferential for the norm on an arbitrary H space H . For this purpose, we exploit that the dual space (H ⊗ π,sym H ) * is isometrically isomorphic to the space of self-adjoint, bounded linear operators. As mentioned above, the action of a specific self-adjoint, bounded linear operator Φ on an arbitrary symmetric tensor w in H ⊗ π,sym H is given by the lifting of the quadratic map u → Φ[u], u H with u ∈ H . In the following, we use the dual pairing notation Φ, w H ⊗ π,sym H to refer to this action. With this preliminary considerations, the dilinear subdifferential of an arbitrary H norm is given in the following manner, where the operator Id denotes the identity and S − (H ) the set of all self-adjoint and negative semi-definite operators on H .
Theorem . (Dilinear Subdifferential of H norms). Let H be a real H space with inner product ·, · H and norm || · || H . The dilinear subdifferential of the squared H norm is given by and hence the dilinear subdifferential in the assertion.
Similarly to the dilinear subdifferential of the squared H norm, we use the identification of the dual space (L C ([ , ]) ⊗ π,sym L C ([ , ])) * with the space of self-adjoint, bounded linear operators to describe the range of the adjoint ( ,Ȃ * k ) T of the dilinearly lifted autoconvolution. More precisely, using the unique identification ( -), and incorporating the factorizationȂ * k =⊙ * • I * k , we may write the range of the adjoint lifted operator as where the self-adjoint, bounded integral operator Φ I * k [ϕ ] is given by In our specific setting, the operator Φ I * k [ϕ ] is additionally compact since the range of⊙ * is contained in the injective tensor product as discussed in the proof of Lemma . . Comparing the range of ( ,Ȃ * k ) T and the subdifferential of the squared H norm in Theorem . , we notice that an element (− Tu † , Id +T ) of ∂ β || · || (u † ) is contained in the range of the adjoint ( ,Ȃ * k ) T if and only if for some ϕ in L C ([ , ]). Since T is a self-adjoint, negative semi-definite operator, the spectrum σ (Φ I * k [ϕ ] ) = σ (Id +T ) of eigenvalues of Φ I * k [ϕ ] is bounded from above by one. Considering that for every eigenfunction related to the eigenvalue λ, we see that the spectrum of is moreover symmetric, which means yields that u † has to be an eigenfunction of Φ I * k [ϕ ] with respect to the eigenvalue one. and the data fidelity term by Although Corollary . gives a convergence rate for the deautoconvolution problem, the employed B distance is usually very weak. Knowing that, besides the true solution u † , the function −u † also solves the deautoconvolution problem, we notice that the B distance ∆ β,( ,Φ I * k [ϕ ] ) cannot measure distances in direction of u † . Since the squared H norm is itself a dilinear functional, in the worst case, it can happen that the B distance is constantly zero, and that the derived convergence rate is completely useless. Depending on the integral operator Φ I * k [ϕ ] or, more precisely, on the eigenspace E of the eigenvalue one, we can estimate the B distance from below in the following manner.
where P E ⊥ denotes the orthogonal projection onto the orthogonal complement of the eigenspace E .
Proof. The self-adjoint, compact integral operator ) possesses a symmetric spectrum, where the eigenspace of the eigenvalue −λ n is given by iE n , see ( ); hence, the spectral theorem implies that the action of Φ I * k [ϕ ] is given by where P E n and P iE n denote the projections onto the eigenspace E n and iE n respectively. Considering that u † is an eigenfunction in E , we may write the B distance for the squared H norm as Using the spectral representation of Φ I * k [ϕ ] , and denoting the kernel of Φ I * k [ϕ ] by E , we have the estimation which yields the assertion.
Remark . . The quality of the convergence rate in Corollary . thus crucially depends on the integral operator Φ I * k [ϕ ] of the source condition. If the true solution u † is the only eigenfunction to the eigenvalue one, the minimizers u δ α of the regularized perturbed problem converges nearly strongly to the true solution u † or −u † or, more precisely, converges to the R-linear subspace spanned by u † in norm topology. If we choose α ∼ δ and assume dim E = , then u δ α converges in L C ([ , ]) / span{u † } with a rate of O(δ / ) strongly.

. Numerical simulations
Besides the theoretical regularization analysis in the previous sections, we now perform numerical experiments to verify the established convergence rates and to examine the error that is not covered by the B distance. As before, we restrict ourselves to the deautoconvolution problem and, moreover, to the 'kernelless' setup with the trivial kernel k ≡ .

. . Construction of valid source elements
In a first step, we numerically construct signals that satisfy the source condition in Theorem . . For this, we approximate the integral operator Φ I * k [ϕ ] : L C ([ , ]) → L C ([ , ]) by applying the midpoint rule. More precisely, we assume for an appropriate large positive integer N . Starting from the source element ϕ, we now determine the eigenfunction u to the major eigenvalue λ . Exploiting the convolution representation of Φ I * k [ϕ ] in ( ), we may compute the action of the integral operator Φ I * k [ϕ ] efficiently by applying the F transform. The eigenfunction u itself can be determined approximately by using the power iteration. To overcome the issue that the spectrum of Φ I * k [ϕ ] is symmetric, which means that −λ is also an eigenvalue, see ( ), we apply the power iteration to the operator Φ I * k [ϕ ] •Φ I * k [ϕ ] . In so doing, we obtain a vector in the span of E ∪ iE , where E is the eigenspace with respect to the eigenvalue λ , and iE with respect to −λ , cf. ( ). The projections to E and iE can, however, simply be computed by For the numerical experiments, we choose the signals ϕ (j) ≔ |ϕ (j) |e i arg(ϕ (j ) ) with Here the indicator function 1 [t ,t ] (t) is if t is contained in the interval [t , t ] and else. Rescaling the source element with respect to the major eigenvalue λ (j) by /λ (j ) ϕ (j) , we easily obtain norm-minimizing solutions u † ≔ u (j) which satisfy the required source condition ( -). The source elements ϕ (j) and the related eigenfunctions u (j) are presented in Figure . The results of the simulations here look quite promising in the sense that the class of functions u † satisfying ( -) is rich on naturally occurring signals.

. . Validation of the theoretical converence rate
To verify the convergence rate numerically, we have to solve the deautoconvolution problem for different noise levels. Referring to [Ger ] and [GHB + ], we here apply the G -N method to the discretized problem with forward operator

Figure :
Numerical construction of norm-minimizing solutions u † ≔ u (j) satisfying the source condition in Proposition . on the basis of an explicitly known source element ϕ (j) . For the approximation of the integral operator Φ I * k [ϕ ] in ( ), a discretization with N = samples has been used. .
Phase of the approximation u * j .

Figure :
Comparison between the norm-minimizing solution u † and the numerically reconstructed signals u δ α with and without regularization. The noise level for u * and u * amounts to δ = . ||u † ||, and for u * to δ = ||u † ||. The regularization for u * and u * corresponds to α = δ . The reconstructionu * is computed without regularization or with α = .
In order to solve the occurring equation systems iteratively, we use the conjugate gradient method with a suitable preconditioner and exploit the T structure of the related system matrix, see for instance [RZR ]. For the exact signal u † arising from the source element ϕ ( ) , the numerical approximations u * j corresponding to the minimizer u δ α of the regularized and non-regularized deautoconvolution problem are shown in Figure . Besides the numerical ill-posedness of the discretized deautoconvolution problem for minor levels δ ≔ || δ − † || of G ian noise, we can here see the smoothing effect of the applied T regularization. Even for considerably higher noise levels like δ = ||u † ||, the reconstruction covers the main features of the unknown normminimizing solution u † . The considered noise level δ here depends on the norm ||u † || of the true signal and not on the norm || † || of the exact data. Depending on our underlying numerical implementation, for the considered signal, the noise level δ = ||u † || corresponds to a G ian noise, whose norm approximately equals || † ||. Unfortunately, the result of the G -N method usually strongly depends on the start value, which have to be a very accurate approximation of the norm-minimizing solution u † for very low noise levels δ . For this reason, we extent the simulations for the convergence rate analysis by choosing randomly created start values around the true solution u † per noise level. Since we have constructed the norm-minimizing solution u † and the exact data † from a specific source element ϕ, besides the convergence rates in Corollary . , we have an explicit upper bound for the B distance between the regularized solution u δ α and the norm-minimizing solution u † as well as for the discrepancy between the forward operator A k [u δ α ] and the perturbed data δ . The convergence rate analysis for the source element ϕ ( ) is presented in Figure . Additionally, Theorem . yields an upper bound for the distance || P E ⊥ (u δ α ) || = ||u δ α − P E (u δ α ) || between u δ α and the ray E spanned by u † . Here all three theoretical convergence rates  and upper bounds match with numerical results. If we consider the discrepancy || A k [u δ α ] − δ || in Figure .c more closely, we notice that the numerical and the theoretical rates coincides except for a multiplicative constant. Consequently, we cannot hope to improve the theoretical rate of O(δ ). The B distance and the distance to the ray spanned by u † have a completely different behaviour. More precisely, we here have regions where the convergence rate is faster and regions where the convergence rate is slower than the theoretical rate of O(δ ). Especially for the distance to the ray, it seems that the overall convergence rate is much faster than the theoretical rate. In some circumstances, our theoretical rate could thus be too pessimistic.
In this instance of the deautoconvolution problem, we can observe that the error || P E (u δ α − u † ) || = ||u δ α − P E (u δ α ) || within the ray E , which is not covered by Corollary . , numerically converges to zero superlinearly with a rate of O(δ ). The shown numerical rate here strongly depends on the starting values of the G -N method, which have been chosen in a small neighbourhood around u † . Choosing start-ing values around −u † , we would observe the same convergence rate to −u † . In fact, the sequence u δ α could be composed of two subsequences -one converging to u † and the other to −u † .

. Conclusion
Starting from the question: how non-linear may a non-linear forward operator be in order to extend the linear regularization theory, we have introduced the classes of dilinear and diconvex mappings, which corresponds to linear, bilinear, and quadratic inverse problems. Exploiting the tensorial structure behind these mappings, we have introduced two different concepts of generalized subgradients and subdifferentials -the dilinear and the representative generalization. We have shown that the classical subdifferential calculus can be partly transferred to both new settings. Although the representative generalization yields stronger computation rules, the related subdifferential unfortunately strongly depends on the non-unique convex representative of the considered diconvex mapping. Besides all differences, there exists several connections between the dilinear and representative subdifferential.
On the basis of these preliminary considerations, we have examined the class of dilinear inverse problems. Analogously to linear inverse problems, the regularizations in the dilinear setting are well posed, stable, and consistent. Using the injective tensor product, which is nearly a predual space of the projective tensor product, as topologizing family, we have seen that the required sequential weak * (semi-)continuity of the forward operator and regularization term may be inherited from the lifted versions. Moreover, we have derived a convergence rate analysis very similar to the linear setting under similar assumptions and requirements. This enables us to give explicit upper bounds for the discrepancy and the B distance between the solution of the regularized problem and the R-minimizing solution.
In a last step, we have applied the developed theory to the deautoconvolution problem that appears in spectroscopy, optics, and stochastics. Although the requirements of the non-linear regularization theory are not fulfilled, our novel approach yields convergence rates based on a suitable range source condition and the dilinear B distance. Depending on the source element, the B distance is here surprisingly strong. In the best case, the solutions of the regularized problems converge strongly to the ray spanned by the true signal, which is the best possible rate with respect to the ambiguities of the problem. Using numerical experiments, we have considered different source elements and the corresponding norm-minimizing solutions, which shows that there exists signals satisfying the required source-wise representation. Finally, we have observed the established error bounds in the numerical simulations.