In this paper we propose a spectral method for learning the following binary latent variable model, shown in Figure 1. The hidden layer, , consists of
. The observed vectorof features is modeled as
where is an unknown weight matrix assumed to be full rank . Here, is the noise level and is an additive noise vector independent of , whose
coordinates are all i.i.d. zero mean and unit variance random variables. For simplicity we assume it is Gaussian, though our method can be modified to handle other noise distributions.
) is the Gaussian-Bernoulli restricted Boltzmann machine (G-RBM) where the distributionis further assumed to have a parametric energy-based structure [25, 15, 50]. G-RBMs were used, e.g., in modeling human motion  and natural image patches .
Given i.i.d. samples from model (2), the goal is to estimate the weight matrix . A common approach for learning is by maximum likelihood. As this function is non-convex, common optimization schemes include the EM algorithm and alternating least squares (ALS). In addition, several works developed iterative methods specialized to G-RBMs [24, 15]. All these methods, however, often lack consistency guarantees and may not be well suited for large datasets due to their potential slow convergence. This is not surprising, as learning under model (2) is believed to be computationally hard; see for example Mossel and Roch .
Over the past years, several works considered variants and specific instances of model (2) under additional assumptions on the distribution or on the weight matrix . For example, when
is a product distribution, the learning problem becomes that of independent component analysis (ICA) with binary signals. In this case, several methods have been derived for estimating and under suitable non-degeneracy conditions were proven to be both computationally efficient and statistically consistent [46, 18, 44, 28, 4, 30]. Similarly, when the hidden units are mutually exclusive, namely has support , the model is a Gaussian mixture (GMM) with spherical components with linearly independent means. Efficient and consistent algorithms have been derived for this case as well [38, 2, 3, 26]. Among those, most relevant to this work are orthogonal tensor decomposition methods . Interestingly, these methods can learn some additional latent models, with hidden units that are not necessarily binary, such as Dirichlet allocation and other correlated topic models .
Learning given the observed data can also be viewed as a noisy matrix factorization problem. If is known to be non-negative, then various non-negative matrix factorization methods can be used. Moreover, under appropriate conditions, some of these methods were proven to be computationally efficient and consistent [17, 6]. For general full rank , the matrix factorization method in Slawski et al.  (SHL) exactly recovers when with a runtime exponential in . This method, however, can handle only low levels of noise and has no consistency guarantees when .
A tensor eigenpair approach
In this paper we propose a novel spectral method for learning which is based on the eigenvectors of both the second order moment matrix and the third order moment tensor of the observed data. We prove that our method is consistent under mild non-degeneracy conditions and achieves the parametric rate for any noise level .
The non-degeneracy conditions we pose are significantly weaker than those required by previous tensor decomposition methods mentioned above. In particular, their assumptions and resulting methods can be viewed as specific cases of our more general approach.
Similarly to the matrix factorization method in Slawski et al. , our algorithm has runtime linear in , polynomial in , and in general exponential in . With our current Matlab implementation, most of the runtime is spent on computing the eigenpairs of a tensor. Practically, our method, implemented without any particular optimization, can learn a model with 12 hidden units in less than ten minutes on a standard PC. Furthermore, the overall runtime can be significantly reduced, since the step of computing the tensor eigenpairs can be embarrassingly parallelized.
In the next section we fix the notation and provide necessary background on tensor eigenpairs. In Section 3 we introduce our method in the case . The case is treated in Section 4. Experiments with our method and comparison to other approaches appear in Section 5. All proofs are deferred to the appendices.
We abbreviate and denote as the -th unit vector with entries . We slightly abuse notation and view a matrix also as the set of its columns, namely is some column of and is the span of all its columns. The unit sphere is denoted by .
A tensor is symmetric if for all permutations of . Here, we consider only symmetric tensors. can also be seen as a multi-linear operator: for matrices with , the tensor-mode product, denoted , is a tensor whose -th entry is
Several types of eigenpairs of a tensor have been proposed. Here, we consider the following definition, termed -eigenpairs by Qi  and -eigenpairs by Lim . Henceforth we just call them eigenpairs.
is an eigenpair of if
Note that if
is an eigenpair then the eigenvalue is simply. In addition, is also an eigenpair. Following common practice, we treat these two pairs as one. So, without loss of generality, we make the convention that .
In contrast to the matrix case, the number of eigenvalues of a tensor can be much larger than . As shown by Cartwright and Sturmfels , for a tensor, there can be at most of them. With precise definitions appearing in Cartwright and Sturmfels , for a generic tensor, all its eigenvalues have multiplicity one and the number of eigenpairs is at most .
In principle, computing the set of all eigenpairs of a general symmetric tensor is a #P problem . Nevertheless, several methods have been proposed for computing at least some eigenpairs, including iterative higher-order power methods [31, 32], homotopy continuation , semidefinite programming , and iterative Newton-based methods [29, 22]. We conclude this section with the definition of Newton-stable eigenpairs  which are most relevant to our work.
Equivalently to (3), eigenpairs of can also be characterized by the function ,
It is easy to verify that a pair with is an eigenpair of if and only if and . The stability of an eigenpair is determined by its Jacobian matrix , more precisely, by its projection into the dimensional subspace orthogonal to . Formally, let be a matrix with orthonormal columns that span the subspace orthogonal to and define the projected Jacobian matrix
An eigenpair of is Newton-stable if the matrix has full rank .
The homotopy continuation method in Chen et al.  is guaranteed to compute all the Newton-stable eigenpairs of a tensor. Alternatively, Newton-stable eigenpairs are attracting fixed points for the iterative orthogonal Newton correction method (O–NCM) in Jaffe et al. . Moreover, O–NCM converges to any Newton-stable eigenpair at a quadratic rate given a sufficiently close initial guess. Finally, for a generic tensor, all its eigenpairs are Newton-stable.
3 Learning in the noiseless case
To motivate our approach for estimating the matrix it is instructive to first consider the ideal noiseless case where . In this case, model (2) takes the form . Our problem then becomes that of factorizing the observed matrix of samples into a product of real and binary low-rank matrices,
To be able to recover we first need conditions under which the decomposition of into and is unique. Clearly, such a factorization can be unique at most up to a permutation of its components; we henceforth ignore this degeneracy. A sufficient condition for uniqueness, similar to the one posed in Slawski et al. , is that is rigid. Formally, is rigid if any non-trivial linear combination of its rows yields a non-binary vector: ,
Condition (7) is satisfied, for example, when the columns of include and for all .
The following proposition, similar in nature to the (affine constrained) uniqueness guarantee in Slawski et al. , shows that under condition (7) the factorization in (6) is unique and fully characterized by the binary constraints.
Let with rigid and full rank with . Let be the unique right pseudo-inverse of so . Then and are unique and for all ,
Hence, under the rigidity condition (7), the matrix factorization problem in (6) is equivalent to the problem of finding the unique set of non-zero vectors that satisfy the binary constraints . The weight matrix is then .
We recover via a two step procedure. First, a finite set of candidate vectors is computed with a guarantee that . Specifically, is computed from the set of eigenpairs of a tensor, constructed from the low order moments of . Typically, the size of will be much larger than , so in the second step is filtered by selecting all that satisfy .
Before describing the two steps in more detail we first state the additional non-degeneracy conditions we pose.
To this end, denote the unknown first, second, and third order moments of the latent binary vector by
p&= E[h] ∈R^d,
We assume the following:
for all .
Condition (I) implies . This in turn implies for all and that at most one variable has . Such an “always on” variable can model a fixed bias to . As far as we know, condition (II) is new and its nature will become clear shortly.
We now describe each step of our algorithm in more detail.
Computing the candidate set
To compute a set that is guaranteed to include the columns of we make use of the second and third order moments of ,
M&= E[x⊗x⊗x]∈R^m×m×m. Given a large number of samples , these can be easily and accurately estimated from the sample . For simplicity, in this section we consider the population setting where , so and are known exactly. and are related to the unknown second and third order moments of in (3) via 
Since both and are full rank, the number of latent units can be deduced by . Since is positive definite, there is a whitening matrix such that
Such a can be computed, for example, by an eigen-decomposition of . Although is not unique, any that satisfies (10) suffices for our purpose. Define the lower dimensional whitened tensor
Denote the set of eigenpairs of by
Our set of candidates is then
The following lemma shows that under condition (I) the set is guaranteed to contain the columns of .
Computing the tensor eigenpairs
By Lemma 1, we may construct a candidate set that contains by first calculating the set of eigenpairs of . Unfortunately, computing the set of all eigenpairs of a general symmetric tensor is computationally hard . Moreover, besides the columns of , the set in (13) may contain many spurious candidates, as the number of eigenpairs of is typically which is much larger than .
Nevertheless, as discussed in Section 2, several methods have been proposed for computing some eigenpairs of a tensor under appropriate stability conditions. The following lemma highlights the importance of condition (II) for the stability of the eigenpairs in . Note that conditions (I)-(II) do not depend on , but only on the distribution of the latent variables .
As suggested by Eq. (8) we select the subset of vectors that satisfy the binary constraints,
Indeed, under condition (I), Proposition 1 implies that and the weight matrix is thus .
Algorithm 1 summarizes our method for estimating in the noiseless case and has the following recovery guarantee.
We note that when and conditions (I)-(II) hold for the empirical latent moments and (rather than and ), Algorithm 1 exactly recovers when and are replaced by their finite sample estimates. The matrix factorization method SHL in Slawski et al.  also exactly recovers in the case . While its runtime is also exponential in , practically it may be much faster than our proposed tensor based approach. This is because SHL constructs a candidate set of size
that can be computed by a suitable linear transformation of thefixed set , as opposed to our candidate set which is constructed by eigenpairs of a tensor. However, SHL does not take advantage of the large number of samples , since only sub-matrices of the sample matrix are used for constructing its candidate set. Indeed, in the noisy case where , SHL has no consistency guarantees and as demonstrated by the simulation results in Section 5 it may fail at high levels of noise. In the next section we derive a robust version of our method that consistently estimates for any noise level .
4 Learning in the presence of noise
The method in Section 3 to estimate is clearly inadequate when . However, we now show that by making several adjustments, the two steps of computing the candidate set and its filtering can be both made robust to noise, yielding a consistent estimator of for any .
Computing the candidate set
As in the case our goal in the first step is to compute a finite candidate set that is guaranteed to contain accurate estimates for the columns of .
To this end, in addition to the second and third order moments and in (3), we also consider the first order moment and define the following
noise corrected moments,
M_σ = & M- σ^2 I_m,
M_σ = & M- σ^2 ∑_i=1^m ( μ⊗e_i ⊗e_i + e_i ⊗μ⊗e_i + e_i ⊗e_i ⊗μ). By assumption, the noise satisfies . Thus, similarly to the moment equations in (9), the modified moments in (4) are related to these of by 
Hence, if and were known exactly, a candidate set that contains could be obtained exactly as in the noiseless case, but with and replaced with and ; namely, first calculate the whitening matrix such that and then compute the eigenpairs of the population whitened tensor
In practice, , , , and are all unknown and need to be estimated from the sample matrix . Assuming , the parameters and can be consistently estimated, for example, by the methods in Kritchman and Nadler . For simplicity, we assume they are known exactly. Similarly, , , are consistently estimated by their empirical means, , , and . So, after computing the plugin estimates such that and , we compute the set of eigenpairs of and for some small take our candidate set as
The following lemma shows that under conditions (I)-(II) the above procedure is stable to small perturbations. Namely, for perturbations of order in and , the method computes a candidate set that contains a subset of vectors that are close to the columns of . Furthermore, these vectors all correspond to Newton-stable eigenpairs of the perturbed tensor and are separated from the other candidates in .
Let be the population quantities in (17) and let be their perturbed versions, inducing the candidate set in (18). If conditions (I)-(II) hold, then there are such that for all the following holds: If the perturbed versions satisfy
then any has a unique such that
Moreover, corresponds to a Newton-stable eigenpair of with eigenvalue and for all ,
The proof is based on the implicit function theorem ; small perturbations to a tensor result in small perturbations to its Newton-stable eigenpairs.
Now, by the delta method, the plugin estimates and
are both close to their population quantities,
∥^K_σ- K_σ∥_F & = O_P(n^-12),
∥^W_σ - W_σ∥_F & = O_P(n^-12). By (3), we have that (19) holds with . Hence, by Lemma 3, the eigenpairs of provide a candidate set that contains vectors that are close to the columns of . In addition, any irrelevant candidate is far away from
. As we show next, these properties ensure that with high probability therelevant candidates can be identified in .
Given the candidate set computed in the first step, our goal now is to find a set of vectors that accurately estimate the columns of . To simplify the theoretical analysis, we assume we are given a fresh sample of size that is independent of . This can be achieved by first splitting a sample of size into two sets of size , one for each step.
Recall that for a vector from model (2) and any
Obviously, when , the filtering procedure in (15) for the noiseless case is inadequate, as typically no will exactly satisfy . Nevertheless, we expect that for a sufficiently small noise level , any that is close to some will result in that is close to being binary, while any sufficiently far from will result in that is far from being binary. A natural measure for how is “far from being binary”, similar to the one used for filtering in Slawski et al. , is simply its deviation from its binary rounding,
Eq. (23) works extremely well for small , but fails for high noise levels. Here we instead propose a filtering procedure based on the classical Kolmogorov-Smirnov goodness of fit test . As we show below, this approach gives consistent estimates of for any .
Before describing the test, we first introduce the probabilistic analogue of the rigidity condition (7). For any , define its corresponding expected binary rounding,
Clearly, and for all . We pose the following expected rigidity condition: for all ,
) is distributed according to the following univariate Gaussian mixture model (GMM),
Denote the cumulative distribution function ofby . For general , this mixture may have up to distinct components. However, for , it reduces to a mixture of two components with means at and . More precisely, for any candidate with corresponding eigenvalue , define the GMM with two components
Denote its cumulative distribution function by . The following lemma shows that under condition (24), fully characterizes the columns of .
Given the empirical candidate set , Lemma 4 suggests ranking all according to their goodness of fit to and taking the candidates with the best fit. More precisely, given a sample that is independent of , for each candidate we compute the empirical cumulative distribution function,
and calculate its Kolmogorov-Smirnov score
Our estimator for is then the set of vectors with the smallest scores . The estimator for is the pseudo-inverse, .
The following lemma shows that for sufficiently large , accurately distinguishes between that are close to the columns of from these that are not.
Let and a sequence of random vectors such that Then,
In contrast, if , then
provided the expected rigidity condition (24) holds.
Algorithm 2 summarizes our method for estimating in the general case where and . The following theorem establishes its consistency.
The runtime of Algorithm 2 is composed of three main parts. First, operations are needed to compute all the relevant moments from the data and to construct the whitened tensor . The most time consuming task is computing the eigenpairs of , which can be done by either the homotopy method or O–NCM. Currently, no runtime guarantees are available for either of these methods. In practice, since there are eigenpairs, these methods spend operations in total. Finally, since there are candidates and each KS test takes operations , the filtering procedure runtime is .
Power-stability and orthogonal decomposition
The exponential runtime of our algorithm stems from the fact that the set of Newton-stable eigenpairs of is typically exponentially large. Indeed, the above algorithm becomes intractable for large values of . However, in some cases, the set of relevant eigenpairs has additional structure so that a smaller candidate set may be computed instead of . Specifically, consider the subset of power-stable eigenpairs of .
An eigenpair is power-stable if its projected Jacobian is either positive or negative definite.
Typically, the number of power-stable eigenpairs is significantly smaller than the number of Newton-stable eigenpairs.222We currently do not know whether the number of power-stable eigenpairs of a generic tensor is polynomial or exponential in . In addition, can be computed by the shifted higher-order power method [31, 32].
Similarly to Lemma 2, one can show that is guaranteed to contain whenever the following stronger version of condition (II) holds: for all , the matrix
is either positive-definite or negative-definite.
As an example, consider the case where has the support . Then model (2) corresponds to a GMM with spherical components with linearly independent means. In this case, both and are diagonal with on their diagonal. Thus, the matrices in (28) take the form which by condition (I) are all negative-definite. In fact, in this case, has an orthogonal decomposition and the orthogonal eigenpairs in are the only negative-definite power-stable eigenpairs of . Similarly, when is a product distribution, the same orthogonal structure appears if the centered moments of are used instead of and . As shown in Anandkumar et al. , the power method, accompanied with a deflation procedure, decompose an orthogonal tensor in polynomial time, thus implying an efficient algorithm in these cases.
We demonstrate our method in two scenarios: (I) simulations from the exact binary model (2); and (II) learning a common population genetic admixture model. Code to reproduce the simulation results can be found at https://github.com/arJaffe/BinaryLatentVariables.
We generated samples from model (2) with hidden units, observable features, and Gaussian noise . The columns of were drawn uniformly from the unit sphere . Fixing a mean vector and a covariance matrix , each hidden vector was generated independently by first drawing and then taking its binary rounding.
Figure 2 shows the error, in Frobenius norm, averaged over independent realizations of as a function of (upper panel) and (lower panel) for five methods: (i) our spectral approach, detailed in Algorithm 2 (Spectral); (ii) Algorithm 2 followed by an additional single weighted least square step detailed in Appendix I (Spectral+WLS); (iii) SHL, the matrix decomposition approach of Slawski et al. 333Code taken from https://sites.google.com/site/slawskimartin/code. For each realization, we aggregated over runs of SHL and chose the output that minimized .; (iv) ALS with a random starting point (see Appendix J); and (v) an oracle estimator that is given the exact matrix and computes via least squares.
As one can see, as opposed to SHL, our method is consistent for and achieves an error rate corresponding to a slope of in the left panel of Fig. 2. In addition, as seen in the right panel of Fig. 2, at low level of noise our method is comparable to SHL, whereas at high level of noise it is far more accurate. Finally, adding a weighted least square step reduces the error for low noise levels, but increases the error for high noise levels.
Population genetic admixture
We present an application of our method to a fundamental problem in population genetics, known as admixture, illustrated in Fig. 3. Admixture refers to the mixing of ancestral populations that were long separated, e.g., due to geographical or cultural barriers [42, 1, 35]. The observed data is an matrix where is the number of modern “admixed” individuals and is the number of relevant locations in their DNA, known as SNPs. Each SNP corresponds to two alleles and different individuals may have different alleles. Fixing a reference allele for each location, takes values in according to the number of reference alleles appearing in the genotype of individual at locus .
Given the genotypes , an important problem in population genetics is to estimate the following two quantities. The allele frequency matrix whose entry is the frequency of the reference allele at locus in ancestral population ; and the admixture proportion matrix whose columns sum to and its entry is the proportion of individual ’s genome that was inherited from population .
A common model for in terms of and is to assume that the number of alleles is the sum of two i.i.d. Bernoulli random variables with success probability . Namely,
Note that under this model
), there are two main differences; the noise is not normally distributed and the matrixis non-binary. Yet, model (2) is expected to be a good approximation whenever various alleles are rare in some populations but abundant in others. Specifically, for ancestral populations that have been long separated, some alleles may become fixed in one population (i.e., reach frequency of 1) while being totally absent in others.
We followed a standard simulation scheme appearing, for example, in Xue et al. , Gravel , Price et al. . First, using SCRM , we simulated ancestral populations separated for generations and generated the genomes of individuals for each. was then computed as the frequency of the reference alleles in each population. Next, the columns of were sampled from a symmetric Dirichlet distribution with parameter . Finally, the genomes of admixed individuals were generated as mosaics of genomic segments of individuals from the ancestral populations with proportions . The mosaic nature of the admixed genomes is an important realistic detail, due to the linkage (correlation) between SNPs . A detailed description is in Appendix K.
We compare our algorithm to two methods. The first is Admixture , one of the most widely used algorithms in population genetics, which aims to maximize the likelihood of . A recent spectral approach is ALStructure , where an estimation of via Chen and Storey  is followed by constrained ALS iterations of and . For our method, two modification are needed for Algorithm 2. First, since the distribution of is not Gaussian, the corrected moments as calculated by (4) do not satisfy (16). Instead, we implemented a matrix completion algorithm derived in  for a similar setup, see Appendix H for more details. In addition, the filtering process described in Section 4 is no longer valid. However, as is relatively small, we were able to perform exhaustive search over all candidate subsets of size and choose the one that maximized the likelihood.
Figure 4 compares the results of the methods for . The spectral method outperforms Admixture and ALStructure for and performs similarly to Admixture for .
This research was funded in part by NIH Grant 1R01HG008383-01A1.
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Appendix A Proof of Proposition 1
So any satisfies the binary constraint . For the other direction, let be such that . Since , the rigidity condition (7) implies . Since is full rank and , must be a column of .
Appendix B Proof of Lemma 1
Since the vector is binary, its second and third order moments are related as follows. For all ,
Since is full rank, . Hence, applying multi-linearly on the moment equations in (9) we obtain
Thus, the equality in (30) is equivalent to
Let be the full rank matrix that satisfies where is the whitening matrix in (10). Then,
Inserting these into (31), the matrix must satisfy
Let be an arbitrary symmetric tensor. Then, a matrix of rank satisfies (33) if and only if for all , , where are eigenpairs of with linearly independent .
By Lemma 6, the set of scaled eigenpairs of is guaranteed to contain the columns of . Since , the set is guaranteed to contain .
To show that each