Abstract
In this work, we conduct a joint analysis of both Vector Space and Language Models for IR using the mathematical framework of Quantum Theory. We shed light on how both models allocate the space of density matrices. A density matrix is shown to be a general representational tool capable of leveraging capabilities of both VSM and LM representations thus paving the way for a new generation of retrieval models. We analyze the possible implications suggested by our findings.
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Notes
- 1.
The Dirac notation establishes that \(|u\rangle \) denotes a unit norm vector in \(\mathbb {H}^n\) and \(\langle u| \) its conjugate transpose.
- 2.
In a more general formulation of the theory, a quantum probability measure reduces to a classical probability measure for any set \(\mathcal {M} = \{M_i\}\) of positive operators \(M_i\) such that \(\sum _i M_i = I_n\). The set \(\mathcal {M}\) is called Positive-Operator Valued Measure (POVM) [12]. Therefore, the properties reported in this paper which apply to a complete set of mutually orthogonal projectors equally hold for a general POVM.
- 3.
In general, the dyads in the mixture don’t need to be orthogonal. However, in this case, the coefficients \(\upsilon _i\) cannot be easily interpreted as the probabilities assigned by the density matrix to each dyad.
- 4.
In quantum physics, the meaning of i.i.d. can be associated to the physical notion of measurement. If a density matrix \(\rho \) represents the state of a system, an i.i.d. set of \(m\) quantum events is obtained by performing a measurement on \(m\) different copies of \(\rho \) and by recording the outcomes.
- 5.
In this paper, we do not explicitly take into account situations in which the vectors could contain negative entries. For example, this could easily happen after the application of Rocchio’s algorithm [16] in feedback situations or by reducing the dimensionality of the vector space by LSI [3]. Besides the historically encountered difficulties in the interpretation of such negative entries [6], in these particular cases, the rank equivalence situations discussed here could not hold. However, we argue that ignoring these situations causes no harm to the generality of our conclusions on the need of an enlarged representation space.
- 6.
This is indeed the practice of Query Expansion (QE), see for example [2].
- 7.
In [8], each basis of a vector space is considered as describing a contextual property and the vectors in the basis as contextual factors. We prefer not to adopt such interpretation for two reasons: (1) in this paper, classical sample spaces are exclusively associated to orthonormal basis and (2) we believe that referring to concepts leads to a more general formulation, better tailored to our needs.
References
Birkhoff, G., Von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823 (1936)
Carpineto, C., Romano, G.: A survey of automatic query expansion in information retrieval. ACM Comput. Surv. 44(1), 1:1–1:50 (2012)
Deerwester, S., Dumais, S.T., Furnas, G.W., Landauer, T.K., Harshman, R.: Indexing by latent semantic analysis. J. Am. Soc. Inf. Sci. 41, 391–407 (1990)
Gao, J., Nie, J.Y., Wu, G., Cao, G.: Dependence language model for information retrieval. In: Proceedings of SIGIR, pp. 170–177 (2004)
Gleason, A.: Measures on the closed subspaces of a hilbert space. J. Math. Mech. 6, 885–893 (1957)
Hofmann, T.: Unsupervised learning by probabilistic latent semantic analysis. Mach. Learn. 42(1), 177–196 (2001)
Lvovsky, A.I.: Iterative maximum-likelihood reconstruction in quantum homodyne tomography. J. Opt. B: Quant. Semiclassical Opt. 6, S556–S559 (2004)
Melucci, M.: A basis for information retrieval in context. ACM Trans. Inf. Syst. 26, 14:1–14:41 (2008)
Melucci, M.: An investigation of quantum interference in information retrieval. In: Cunningham, H., Hanbury, A., Rüger, S. (eds.) IRFC 2010. LNCS, vol. 6107, pp. 136–151. Springer, Heidelberg (2010)
Melucci, M.: Deriving a quantum information retrieval basis. The Computer Journal 56(11), 1279–1291 (2013). doi:10.1093/comjnl/bxs095
Melucci, M., Rijsbergen, K.: Quantum mechanics and information retrieval. Adv. Top. Inf. Retrieval 33, 125–155 (2011). (Springer, Berlin)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2004)
Piwowarski, B., Frommholz, I., Lalmas, M., van Rijsbergen, K.: What can quantum theory bring to information retrieval. In: Proceedings of CIKM, pp. 59–68 (2010)
Piwowarski, B., Amini, M.R., Lalmas, M.: On using a quantum physics formalism for multidocument summarization. JASIST 63(5), 865–888 (2012)
van Rijsbergen, K.: The Geometry of Information Retrieval. Cambridge University Press, Cambridge (2004)
Rocchio, J.: Relevance feedback in information retrieval. In: Salton, G. (ed.) The SMART Retrieval System, pp. 313–323. Prentice-Hall, Englewood Cliffs (1971)
Salton, G., Buckley, C.: Term-weighting approaches in automatic text retrieval. Inf. Process. Manage. 24(5), 513–523 (1988)
Song, D., Lalmas, M., van Rijsbergen, K., Frommholz, I., Piwowarski, B., Wang, J., Zhang, P., Zuccon, G., Bruza, P., Arafat, S., Azzopardi, L., Huertas-Rosero, A., Hou, Y., Melucci, M., Rüger, S.: How quantum theory is developing the field of information retrieval. In: Proceedings of QI, pp. 105–108 (2010)
Sordoni, A., Nie, J.-Y., Bengio, Y.: Modeling term dependencies with quantum language models for IR. In: Proceedings of SIGIR, pp. 653–662 (2013)
Sordoni, A., He, J., Nie, J.-Y.: Modeling latent topic interactions with quantum interference for IR. To appear in Proceedings of CIKM (2013)
Tsuda, K., Ratsch, G., Warmuth, M.K.: Matrix exponentiated gradient updates for on-line learning and bregman projection. J. Mach. Learn. Res. 6(1), 995 (2006)
Warmuth, M.K., Kuzmin, D.: Bayesian generalized probability calculus for density matrices. Mach. Learn. 78(1–2), 63–101 (2009)
Widdows, D., Peters, S.: Quantum logic of word meanings: Concept lattices in vector space models (2003)
Wong, S.K.M., Yao, Y.Y.: On modeling information retrieval with probabilistic inference. ACM Trans. Inf. Syst. 13(1), 38–68 (1995)
Zhai, ChX: Statistical language models for information retrieval a critical review. Found. Trends Inf. Retr. 2(3), 137–213 (2007)
Zhao, X., Zhang, P., Song, D., Hou, Y.: A novel re-ranking approach inspired by quantum measurement. In: Clough, P., Foley, C., Gurrin, C., Jones, G.J.F., Kraaij, W., Lee, H., Mudoch, V. (eds.) ECIR 2011. LNCS, vol. 6611, pp. 721–724. Springer, Heidelberg (2011)
Zobel, J., Moffat, A.: Exploring the similarity space. SIGIR Forum 32(1), 18–34 (1998)
Zuccon, G., Azzopardi, L.: Using the quantum probability ranking principle to rank interdependent documents. In: Gurrin, C., He, Y., Kazai, G., Kruschwitz, U., Little, S., Roelleke, T., Rüger, S., van Rijsbergen, K. (eds.) ECIR 2010. LNCS, vol. 5993, pp. 357–369. Springer, Heidelberg (2010)
Zuccon, G., Piwowarski, B., Azzopardi, L.: On the use of complex numbers in quantum models for information retrieval. In: Amati, G., Crestani, F. (eds.) ICTIR 2011. LNCS, vol. 6931, pp. 346–350. Springer, Heidelberg (2011)
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Sordoni, A., Nie, JY. (2014). Looking at Vector Space and Language Models for IR Using Density Matrices. In: Atmanspacher, H., Haven, E., Kitto, K., Raine, D. (eds) Quantum Interaction. QI 2013. Lecture Notes in Computer Science(), vol 8369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54943-4_13
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