Quantum Latent Semantic Analysis

TL;DR

This paper introduces a quantum-inspired latent semantic analysis method that combines geometric and probabilistic approaches, showing promising results on standard datasets for document analysis.

Contribution

It proposes a novel quantum-based latent topic analysis technique that integrates geometry and probability within a unified framework.

Findings

Outperforms LSA on two of three datasets

Supports a geometrical and probabilistic interpretation

Encourages further exploration of quantum-inspired models

Abstract

The main goal of this paper is to explore latent topic analysis (LTA), in the context of quantum information retrieval. LTA is a valuable technique for document analysis and representation, which has been extensively used in information retrieval and machine learning. Different LTA techniques have been proposed, some based on geometrical modeling (such as latent semantic analysis, LSA) and others based on a strong statistical foundation. However, these two different approaches are not usually mixed. Quantum information retrieval has the remarkable virtue of combining both geometry and probability in a common principled framework. We built on this quantum framework to propose a new LTA method, which has a clear geometrical motivation but also supports a well-founded probabilistic interpretation. An initial exploratory experimentation was performed on three standard data sets. The results show that the proposed method outperforms LSA on two of the three datasets. These results suggests that the quantum-motivated representation is an alternative for geometrical latent topic modeling worthy of further exploration.

Figures

Tables

Table 1. Table 1: Summary of the retrieval performance on the test collections. Reported values are Mean Average Precision over all the available queries.

	MED		CRAN		CACM
Method	Precision	Improv.	Precision	Improv.	Precision	Improv.
cosine	0.4678	-	0.2809	-	0.1722	-
LSA	0.5069	+8.36%	0.3302	+17.55%	0.1630	-5.34%
QLSA	0.5443	+16.35%	0.3504	+24.74%	0.1315	-23.64%

Equations

$${\phi }_i(j)=\sqrt{\frac{t{d}_{ji}}{{\sum }_{j=1}^{m}t{d}_{ji}}}\text{, for all }j=1\dots m,$$

$$\forall i,\Vert {\phi }_i\Vert =1$$

$$<{\phi }_i,{\tau }_j{>}^2=P(t_j|d_i)$$

$$P(P_{{\tau }_j}|{\rho }_i)=\text{tr}({\rho }_i{P}_{{\tau }_j})=\text{tr}(|{\phi }_i⟩⟨{\phi }_i|{\tau }_j⟩⟨{\tau }_j|)=⟨{\phi }_i|{\tau }_j⟩{}^2,$$

$$P_S=\sum _{k=1}^r|{\sigma }_k⟩⟨{\sigma }_k|,$$

$$|\overline{{\phi }_i}⟩=P_S|{\phi }_i⟩.$$

$$P(z_k|d_i)={⟨\widehat{{\phi }_i}|{\sigma }_k⟩}^2=\frac{{⟨{\phi }_i|{\sigma }_k⟩}^2}{{\Vert P_S|{\phi }_i⟩\Vert }^2}.$$

$$\underset{\genfrac{}{}{0ex}{}{S}{dim(S)=r}}{\min }\sum _{i=1}^n{\Vert |\overline{{\phi }_i}⟩-|{\phi }_i⟩\Vert }^2=\underset{\genfrac{}{}{0ex}{}{S}{dim(S)=r}}{\min }\sum _{i=1}^n{\Vert P_S|{\phi }_i⟩-|{\phi }_i⟩\Vert }^2$$

$$\Phi =U\Sigma V^T,$$

$${\Phi }_{ij}=\sqrt{\frac{t{d}_{ji}}{{\sum }_{j=1}^{m}t{d}_{ji}}}$$

$$|\overline{{\phi }_i}⟩=\sum _{k=1}^r|{\sigma }_k⟩⟨{\sigma }_k||{\phi }_i⟩$$

$$|\overline{{\phi }_i}⟩=\frac{|\overline{{\phi }_i}⟩}{\Vert |\overline{{\phi }_i}⟩\Vert }$$

$$P(t_j|d_i)=\overline{{\phi }_i}(j{)}^2$$

$$P(z_k|d_i)={⟨\overline{{\phi }_i}|{\sigma }_k⟩}^2$$

$$P(t_j|d_i)=\sum _{k=1}^{r}P(t_j|z_k)P(z_k|d_i),$$

$${\phi }_i(j)$$

$$P(t_j|d_i)$$

Full text

1 Introduction

Since its inception, latent topic analysis111For the remaining part of the text we will use the term latent topic analysis to allude the general modeling strategy avoiding confusion with latent semantic analysis which refers to the particular method. (LTA) (also known as latent semantic analysis/indexing) has been a valuable technique for document analysis and representation in both information retrieval (IR) and machine learning. The main assumption behind LTA is that the observed term-document association may be explained by an underlying latent topic structure. Different methods for latent topic analysis have been proposed, the most prominent include: latent semantic analysis (LSA) [2], probabilistic latent semantic analysis (PLSA) [4], and latent Dirichlet allocation (LDA) [1]. LSA was the first latent analysis method proposed and its approach is geometrical in nature, while PLSA and LDA have a sound probabilistic foundation.

Quantum information retrieval (QIR) [12, 10], is a relatively new research area that attempts to provide a foundation for information retrieval building on the mathematical framework that supports the formulation of quantum mechanics (QM). QIR assimilates the traditional vector space representation to Hilbert spaces, the fundamental concept in QM. Notions such as system state, measurement, uncertainty and superposition are interpreted in the context of IR. QIR is been actively researched and some results suggest that it can go beyond an interesting analogy to become a valuable theoretical and methodological framework for IR [10].

The main goal of this paper is to explore latent topic analysis in the context of QIR. Same as in the vector space model, QIR represents documents/queries as vectors in a vector space (more precisely, a Hilbert space), however, QIR exploits the subspace structure of the Hilbert space and corresponding probability measures to define important IR notions, such as relevance, in a principled way [12]. A question that emerges is whether the richer QIR representation could provide new insights into the latent topic analysis problem. One important motivation for this question is the fact that QIR naturally combines both geometry and probability. Latent topic analysis methods proposed so far are either geometrical or probabilistic in nature, but not both. A quantum-motivated latent semantic analysis method could potentially combine both perspectives.

Some works in QIR [3, 9, 13, 8] have already suggested the relationship between LTA and a quantum-based representation of documents. Up to our knowledge, there has not been proposed yet an original LTA algorithm in a quantum representation context. The work of Melucci [8] probably is the closest one to the work presented in this paper. In that work, a framework for modeling contexts in information retrieval is presented. The framework uses both a quantum representation of documents and LSA to model latent contexts, but do not propose a new LTA method.

This paper proposes a new LTA method, quantum latent semantic analysis (QLSA). The method starts from a quantum-motivated representation of a document set in a Hilbert space $H$. The latent topic space is modeled as a sub-space of $H$, where the document set is projected. The method is analyzed from geometrical and probabilistic points of view, and compared with LSA and PLSA. An exploratory experimentation was performed to evaluate how the quantum-motivated representation impacts the performance of the method. The results show that the method outperforms LSA on two of the three datasets, and we hypothesize that it is due to an improved quantum representation.

The paper is organized as follows: Section 2 provides a brief overview of quantum information retrieval; Section 3 describes the method and discusses its similarities and differences with LSA and PLSA; Section 3 covers the exploratory experimental evaluation of the method; finally, Section 4 presents some conclusions and the future work.

2 Quantum Information Retrieval

QIR provides an alternative foundation for information retrieval. The main ideas were initially proposed by Van Rijsbergen [12], and different subsequent works have contributed to the continuous development of the area. The main idea in QIR is to use the quantum mechanics formalism to deal with fundamental information retrieval concepts exploiting clear analogies between both areas. For instance, a quantum system state is represented by a wave function, which can be seem as a finite or infinite complex vector indexed by a continuous or discrete variable (usually representing space or momentum). In a vector space model, documents are represented by vectors, but in this case they are finite real vectors indexed by a discrete variable that represents text terms. In the next paragraphs we will briefly present some basic concepts from QIR that are necessary to introduce the proposed method.

Lets $D=\{d_{i}\}_{i=1\dots n}$ be a set of documents, $T=\{t_{j}\}_{j=1\dots m}$ be a set of terms, and $TD=\{td_{ji}\}$ be the corresponding term-document matrix. The quantum representation of a document $d_{i}$ is given by a wave function $\varphi_{i}$ defined by:

[TABLE]

This representation has the following convenient properties:

[TABLE]

[TABLE]

where $<\cdot,\cdot>$ is the dot product operator, $\tau_{j}$ is the wave function of the term $t_{j}$ corresponding to a unitary vector with a one in the $j$-th position. This representation corresponds in fact to a representation of the documents in the term space, which we will call $H$ and whose basis is $\{\left|\tau_{j}\right\rangle\}_{j=1\dots m}$.

Dirac notation is a convenient notation formalism extensively used in quantum mechanics. The two basic building blocks of Dirac notation are the bra and the ket, notated respectively as $\left\langle\varphi\right|$ and $\left|\beta\right\rangle$. A ket represents a vector in a Hilbert space and a bra a function from the Hilbert space to a real (or complex) space. The application of a bra to a ket coincides with the dot product of the corresponding vectors and is notated $\left\langle\varphi|\beta\right\rangle$. In a finite-dimensional Hilbert space, a bra may be seem as a row vector and a ket as a column vector, in this case the application of a bra to a ket would correspond to a conventional matrix multiplication.

A bra and a ket can be composed in a reverse way, $\left|\beta\right\rangle\left\langle\varphi\right|$, and this can be interpreted as the outer product of the corresponding vectors. This is useful, for instance, to define notions such as subspace projectors. A subspace is determined by a basis that generates it or by a projector operator that projects any vector in the space to the subspace. If the basis of a given subspace $S$ is $\{\beta_{1},\dots,\beta_{m}\}$, the corresponding projector is $P_{s}=\sum_{i=1\dots m}\left|\beta_{i}\right\rangle\left\langle\beta_{i}\right|$. Projectors with trace one are called density operators and have an important role in quantum mechanics, they are used to represent the statistical state of a quantum system.

Using Dirac notation the second property in Eq. 1 can be expressed as $\left\langle\varphi_{i}|\tau_{j}\right\rangle{}^{2}=P(t_{j}|d_{i})$. This property can be interpreted, in a QIR context, as the density operator $\rho_{i}=\left|\varphi_{i}\right\rangle\left\langle\varphi_{i}\right|$ (corresponding to the document $d_{i}$) acting on the subspace $P_{\tau_{j}}=\left|\tau_{j}\right\rangle\left\langle\tau_{j}\right|$ (which is induced by the term $t_{j}$) according to the rule:

[TABLE]

where $tr(\cdot)$ is the matrix trace operator. The above procedure could be extended to more complex subspaces, i.e., with dimension higher than one.

3 Quantum Latent Semantic Analysis

In general, LTA modeling assumes that the high diversity of terms in a set of documents may be explained by the presence or absence of latent semantic topics in each document. This induces a new document representation where documents are projected to a latent topic space by calculating the relative degree of presence of each topic in each document. Since the set of latent semantic topics is usually one or two orders of magnitude smaller than the set of terms, the effective dimension of the latent topic space is smaller than the dimension of the original space, and the projection of the document to it is, in fact, a dimensionality reduction process.

A latent topic space is a subspace $S$ of $H$ defined implicitly by its projector as:

[TABLE]

where $\{\left|\sigma_{k}\right\rangle\}_{k=1\dots r}$ is an orthonormal basis of the sub-space $S$ and each $\left|\sigma_{k}\right\rangle$ corresponds to the wave function of a latent topic $z_{k}$. A projection of a document represented by $\left|\varphi_{i}\right\rangle$ on the latent space is given by:

[TABLE]

From a quantum mechanics perspective, this projection can be interpreted as the measurement of the observable corresponding to $S$ on the system state $\left|\varphi_{i}\right\rangle$. This measurement will make the state of the system collapse to a new state $\left|\hat{\varphi_{i}}\right\rangle=\frac{\left|\bar{\varphi_{i}}\right\rangle}{\left\|\left|\bar{\varphi_{i}}\right\rangle\right\|}$. Accordingly, the conditional probability of latent topic $z_{k}$ given a document $d_{i}$ represented in the latent space can be calculated by:

[TABLE]

Now, the main problem is to find an appropriate latent semantic topic space $S$. This can be accomplished by imposing some conditions. In particular, we expect that the latent topic representation loses as few information as possible and be as compact as possible. This can be expressed trough the following optimization problem:

[TABLE]

This problem is solved by performing a singular value decomposition (SVD) on the matrix formed by the vectors corresponding to the wave functions of the documents in the document set. Specifically, a matrix where the $i$-th column corresponds to the ket $\left|\varphi_{i}\right\rangle$, $\Phi=\left[\varphi_{1}\dots\varphi_{n}\right]$, with

[TABLE]

its SVD decomposition. The columns of $U=\left[\sigma_{1}\dots\sigma_{r}\right]$, correspond to the vectors of an orthonormal basis of the latent subspace $S$. The process is summarized in Algorithm 1.

3.1 QLSA vs LSA

Both QLSA and LSA use SVD as the fundamental method to find the latent space. However, there is an important difference: LSA performs the SVD decomposition of the original term-document matrix, whereas QLSA decomposes the document wave function matrix, whose entries are proportional to the square root of the original term-document matrix. This makes QLSA a different method, since the decomposition is happening on a different representation space.

Both methods have a clear geometrical motivation, however QLSA has, in addition, a natural probabilistic interpretation. LSA produces a representation that may include negative values, this has been pointed as a negative characteristic of latent topic representations based on SVD [7, 14], since a document may be represented by both the presence and the absence of terms or topics in it. QLSA, in contrast, always produces positive values when documents are mapped back to the term/topic space.

3.2 QLSA vs PLSA

The approach followed by PLSA is quite different to the one of QLSA. PLSA has a strong statistical foundation that models documents as a mixture of term probabilities conditioned on a latent random variable [4]. The parameters of the model are estimated by a likelihood maximization process based on expectation maximization. The mixture calculated by PLSA induces a factorization of the original term-document matrix:

[TABLE]

where $P(t_{j}|z_{k})$ codifies the latent topic vectors and $P(z_{k}|d_{i})$ corresponds to the representation of documents on the latent space.

QLSA also induces a factorization, but of the matrix formed by the wave functions corresponding to the documents in the set. To illustrate this lets check how the wave function of a document $d_{i}$ is codified by QLSA:

[TABLE]

Eq. 3 induces a factorization of the document wave function matrix $\Phi$ into two matrices, one codifying the latent topic wave functions $\left|\sigma_{k}\right\rangle$ represented in the term space, and the other one representing the interaction between documents and latent topics.

Using 1 and 3 we can calculate the approximation of $P(t_{j}|d_{i})$ generated by QLSA:

[TABLE]

where $I_{ji}=\left[\sum_{k,l=1\dots r,k\neq l}\left\langle\tau_{j}|\sigma_{k}\right\rangle\left\langle\sigma_{k}|\varphi_{i}\right\rangle\left\langle\tau_{j}|\sigma_{l}\right\rangle\left\langle\sigma_{l}|\varphi_{i}\right\rangle\right]/\left\|P_{S}\left|\varphi_{i}\right\rangle\right\|^{2}$. Checking 2 and 4 it is easy to see the difference between both approximations, QLSA adds the additional term $I_{ji}$. This term could be interpreted as an interference term [15].

4 Experimental Evaluation

In this section we perform an exploratory experimentation that evaluates the performance of QLSA against LSA. As discussed in Section 3.1, both methods share a common geometrical approach that finds a low-dimensional space using SVD. The main difference resides in the document representation used. Thus, the goal of the experimental evaluation is to establish the effect of the quantum representation when using a latent topic indexing strategy for document retrieval.

In our experiments we evaluated the automatic indexing task to support query based retrieval. The performance is measured in terms of Mean Average Precision (MAP) for two standard datasets to assess the empirical differences between the formulated method and two baseline approaches: direct matching in a Vector Space Model, using cosine similarity, and the LSA approach. The experimental setup is intentionally kept simple, only term frequency is used without any kind of weighting, simple stop-word removal and stemming preprocessing is applied. Document search is performed by projecting the query terms and using the cosine similarity with respect to other documents in the latent space, i.e., ranking scores are taken directly from the latent space.

4.1 Collections

To follow an evaluation of ranked retrieval, we used three collections with relevance assessment: (1) the MED collection, a common dataset used in early information retrieval evaluation, composed of 1033 medical abstracts and 30 queries, all indexed with about 7000 terms; (2) the CRAN collection, another standard dataset with 1400 document abstracts on aeronautics from the Cranfield institute of Technology and 225 queries, is indexed with about 3700 terms. (3) The CACM collection, with 3204 abstracts from the Communications of the ACM Journal with 64 queries, is indexed with about 3000 terms.

4.2 Dimensions of the Latent Space

Figure 1 presents the variation of MAP with respect to the number of latent factors for the evaluated collections. It shows that latent indexing methods provide an improvement over the cosine similarity baseline for the MED and CRAN collections. The dimension of the latent space was varied from 50 to 300 factors taking steps of 10 units for the MED collection and from 100 to 1000 factors taking steps of 50 units for the CRAN collection. The CACM collection, however, does not show improvements when using latent factors for document indexing.

For the first two collections, results show that QLSA performs better than LSA for every evaluated dimension of the latent topic space. In the MED collection, the performance of both methods increases to reach a maximum value around the same latent space dimensionality (between 140 and 160) and then starts to decrease slowly again. In the CRAN collection, the performance of both methods increases and tends to get stable after 500 topics. The best number of topics is very similar for both methods, however, the performance is significantly improved in favor of QLSA.

The CACM collection is particularly challenging for LSA, and QLSA does not perform better. In fact, QLSA seems to amplify the bad performance of LSA. In the case of LSA, this is consistent with previously reported performances in the literature, that showed no benefit for query based retrieval, but instead, a decreasing in performance.

4.3 Recall-Precision Evaluation

Figure 2 shows the interpolated Recall-Precision graph for the 3 evaluated approaches, averaged over the available set of queries. Each model has been configured with the best latent space dimensionality, according to the analysis on the previous Section. Again, results show that latent topic indexing provides a better response over the direct matching approach in the MED and CRAN collections. The plots also show an improved response of QLSA over both cosine and LSA approaches, in these two collections.

In the MED collection, QLSA provides a slightly better response with respect to the cosine similarity in the early stages of the retrieval process, and then starts to show a larger improvement. LSA starts worse than cosine but after the first part of the results it overtakes the baseline and shows a better response in the long term retrieval. QLSA presents a better response than LSA during the whole retrieval process. In the case of the CRAN collection, QLSA and LSA show a general improvement over the baseline, both in the early and long term retrieval. QLSA again offers better results than the other two methods, showing a consistent improvement in terms of precision for the ranked retrieval task.

Figure 2-c shows the response of the indexing methods on the CACM collection, showing an important decreasing for QLSA. We hypothesize that, for this collection, discriminative terms are mixed with other terms in latent factors, leading to a lose of discerning capacity of the ranking method.

Table 1 summarizes the results obtained in this exploratory evaluation, showing that QLSA results in an important improvement with respect to LSA for two collections even though both algorithms are based on a SVD. These results complement the theoretical differences between both algorithms and highlight the empirical benefits of using a QIR-based algorithm for modelling latent topics. In the case of the CACM collection, both LSA and QLSA show a decreasing in performance with respect to the baseline, with a larger margin for QLSA. It is interesting to see that when LSA performs better than the baseline, QLSA is able to outperform both, the baseline and LSA. But, when LSA does not improve, QLSA performs even worse.

A comparison against PLSA was not performed, however, the results reported by [4] could serve as a reference, despite they were obtained with a slightly different experimental setup that favors the performance of the algorithms. It reports an average precision of 63.9, 35.1 and 22.9 for MED, CRAN and CACM respectively, using PLSA. According to these results, QLSA does not outperforms PLSA, however, it shows a competitive performance on two of the datasets, on the other one the performance was remarkable bad.

5 Discussion and Conclusions

Given its exploratory nature, the experimental results are not conclusive. However, the results are encouraging and suggest that the quantum representation could provide a good foundation for latent topic analysis. The approaches followed by both QLSA and LSA are very similar, the main difference is the document representation used. It is interesting to see the effect of the quantum representation on LSA performance: it improved the performance on two of the datasets where LSA showed some advantage over the baseline, but also it amplified the bad performance on the other dataset. However, QLSA has a clear advantage over LSA, its more principled representation of the geometry of the document space allows a probabilistic interpretation.

LTA methods based on probabilistic modeling, such as PLSA and LDA, have shown better performance than geometry-based methods. However, with methods such as QLSA it is possible to bring the geometrical and the probabilistic approaches together. Here we started from a geometrical stand point to formulate the model and then we provided a probabilistic interpretation of it. Thanks to the dual nature of the quantum representation, it is possible to do exactly the opposite: start from a probabilistic latent topic model and then give it a geometrical interpretation. A good start point would be the theory of quantum probabilistic networks [11, 5, 6].

There are many remaining open questions that justify further investigation: what is the interpretation of the interference term (Eq. 4) in the approximation of $P(t_{j}|d_{i})$ generated by QLSA? How to implement quantum versions of probabilistic LTA methods such as PLSA and LDA? These questions are the main focus of our ongoing research work.