We formulate learning of a binary autoencoder as a biconvex optimization problem which learns from the pairwise correlations between encoded and decoded bits. Among all possible algorithms that use this information, ours finds the autoencoder that reconstructs its inputs with worst-case optimal loss. The optimal decoder is a single layer of artificial neurons, emerging entirely from the minimax loss minimization, and with weights learned by convex optimization. All this is reflected in competitive experimental results, demonstrating that binary autoencoding can be done efficiently by conveying information in pairwise correlations in an optimal fashion.
Our approach also isolates the benefit of higher @math in dealing with overfitting, as the pairwise correlations @math can be measured progressively more accurately as @math increases. In this respect, we follow a line of research using such pairwise correlations to model arbitary higher-order structure among visible units, rooted in early work on (restricted) Boltzmann Machines ( @cite_37 @cite_1 @cite_12 @cite_26 ). More recently, theoretical algorithms have been developed with the perspective of learning from the correlations between units in a network, under various assumptions on the activation function, architecture, and weights, for both deep ( @cite_38 ) and shallow networks (using tensor decompositions, e.g. @cite_36 @cite_14 ). Our use of ensemble aggregation techniques (from @cite_18 @cite_28 ) to study these problems is anticipated in spirit by prior work as well, as discussed at length by @cite_25 in the context of distributed representations.
@cite_38 @cite_37 @cite_26 @cite_14 @cite_18 @cite_36 @cite_28 @cite_1 @cite_25 @cite_12
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We report on a novel technique called spatial coupling and its application in the analysis of random constraint satisfaction problems (CSP). Spatial coupling was invented as an engineering construction in the area of error correcting codes where it has resulted in efficient capacity-achieving codes for a wide range of channels. However, this technique is not limited to problems in communications, and can be applied in the much broader context of graphical models. We describe here a general methodology for applying spatial coupling to random constraint satisfaction problems and obtain lower bounds for their (rough) satisfiability threshold. The main idea is to construct a distribution of geometrically structured random K-SAT instances - namely the spatially coupled ensemble - which has the same (rough) satisfiability threshold, and is at the same time algorithmically easier to solve. Then by running well-known algorithms on the spatially coupled ensemble we obtain a lower bound on the (rough) satisfiability threshold of the original ensemble. The method is versatile because one can choose the CSP, there is a certain amount of freedom in the construction of the spatially coupled ensemble, and also in the choice of the algorithm. In this work we focus on random K-SAT but we have also checked that the method is successful for Coloring, NAE-SAT and XOR-SAT. We choose Unit Clause propagation for the algorithm which is analyzed over the spatially coupled instances. For K = 3, for instance, our lower bound is equal to 3.67 which is better than the current bounds in the literature. Similarly, for graph 3-colorability we get a bound of 2.22 which is also better than the current bounds in the literature. Determining interrelatedness structure of various entities from multiple time series data is of significant interest to many areas. Knowledge of such a structure can aid in identifying cause and effect relationships, clustering of similar entities, identification of representative elements and model reduction. The majority of existing results are based on correlation based indices which effectively assume a static relationship between the time series data and are not suitable for detecting interrelatedness when the time series are dynamically related or when the time series involve loops. In this paper, a methodology for identifying the interrelatedness structure of dynamically related time series data is presented that also allows for the presence of loops in the connectivity structure. A linear dynamic graph model is presented where it is assumed that each time series data is the sum of an independent stochastic noise source and a dynamically weighted sum of other time series data. A link is assumed to be present between two time series if the weight of a time series, which is a linear time-invariant filter, is nonzero in the formation of the other. Reconstruction of the link connectivity structure under various scenarios is considered. It is shown that when the linear dynamic graph is allowed to admit non-causal weights, then the links structure can be recovered with the possibility of identifying spurious connections. However, it is shown that the spurious links remain local, where, a spurious link is restricted to be within one hop of a true link. Furthermore, strategies for exact reconstruction of the link structure when the weights are restricted to be causal are developed. The main tools for determining the network topology are based on variations of Wiener filtering. A significant insight provided by the article is that, in the class of network models considered in the paper, the Wiener filter estimating a stochastic process based on other processes remains local in the sense that the Wiener filter utilizes only measurements local to the node being estimated. The aim of this paper is to show that spatial coupling can be viewed not only as a means to build better graphical models, but also as a tool to better understand uncoupled models. The starting point is the observation that some asymptotic properties of graphical models are easier to prove in the case of spatial coupling. In such cases, one can then use the so-called interpolation method to transfer known results for the spatially coupled case to the uncoupled one. Our main use of this framework is for Low-density parity check (LDPC) codes, where we use interpolation to show that the average entropy of the codeword conditioned on the observation is asymptotically the same for spatially coupled as for uncoupled ensembles. We give three applications of this result for a large class of LDPC ensembles. The first one is a proof of the so-called Maxwell construction stating that the MAP threshold is equal to the area threshold of the BP GEXIT curve. The second is a proof of the equality between the BP and MAP GEXIT curves above the MAP threshold. The third application is the intimately related fact that the replica symmetric formula for the conditional entropy in the infinite block length limit is exact. We develop a novel modeling framework for Boltzmann machines, augmenting each hidden unit with a latent transformation assignment variable which describes the selection of the transformed view of the canonical connection weights associated with the unit. This enables the inferences of the model to transform in response to transformed input data in a stable and predictable way, and avoids learning multiple features differing only with respect to the set of transformations. Extending prior work on translation equivariant (convolutional) models, we develop translation and rotation equivariant restricted Boltzmann machines (RBMs) and deep belief nets (DBNs), and demonstrate their effectiveness in learning frequently occurring statistical structure from artificial and natural images. We describe a generative model of the relationship between two images. The model is defined as a factored three-way Boltzmann machine, in which hidden variables collaborate to define the joint correlation matrix for image pairs. Modeling the joint distribution over pairs makes it possible to efficiently match images that are the same according to a learned measure of similarity. We apply the model to several face matching tasks, and show that it learns to represent the input images using task-specific basis functions. Matching performance is superior to previous similar generative models, including recent conditional models of transformations. We also show that the model can be used as a plug-in matching score to perform invariant classification. Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ|I|. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ< λc=(Δ-1)Δ-1 (Δ-2)Δ. This improves on the previously known general bound of λ ≤ (2 ‾ Δ-2). The new regime includes the interesting case of λ=1 (uniform weights) and Δ ≤ 5. The previous bound required Δ ≤ 4 for uniform approximate counting and there is evidence that for Δ ≥ 6 the problem is hard. Note that λc is the critical activity for uniqueness of the Gibbs measure on the regular tree of degree Δ, i.e., for λ ≤ λc the probability that the root is in the independent set is asymptotically independent of the configuration on the leaves far below. Indeed, our analysis is focused on establishing decay of correlations with distance in the above weighted distribution. We show that on any graph of maximum degree Δ correlations decay with distance at least as fast as they do on the regular tree of the same degree. This resolves an open conjecture in statistical physics. Our comparison of a general graph with the tree uses an algorithmic argument yielding the approximation scheme mentioned above. Also, by existing arguments, establishing decay of correlations for all graphs and λ<λc gives that the Glauber dynamics is rapidly mixing in this regime. However, the implication from decay of correlations to rapid mixing of the dynamics is only known to hold for graphs of subexponential growth, and hence, our result regarding the Glauber dynamics is limited to this class of graphs. We study a noiseless broadcast link serving @math users whose requests arise from a library of @math files. Every user is equipped with a cache of size @math files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most @math file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file @math scales to infinity. The asymptotic coding gain obtained is roughly @math . In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets @math is a function of the system parameters @math , and @math . Specifically, we show that the existing random placement and clique cover delivery schemes that achieve optimality in the asymptotic regime can have at most a multiplicative gain of 2 even if the number of packets is exponential in the asymptotic gain @math . Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of @math is at least @math . We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters. Relation learning is a fundamental operation in many computer vision tasks. Recently, high-order Boltzmann machine and its variants have exhibited the great power of modelling various data relation. However, most of them are unsupervised learning models which are not very discriminative and thus cannot server as a standalone solution to relation learning tasks. In this paper, we explore supervised learning algorithms and propose a new model named Conditional High-order Boltzmann Machine (CHBM), which can be directly used as a bilinear classifier to assign similarity scores for pairwise images. Then, to better deal with complex data relation, we propose a gated version of CHBM which untangles factors of variation by exploiting a set of latent variables to gate classification. We perform four-order tensor factorization for parameter reduction, and present two efficient supervised learning algorithms from the perspectives of being generative and discriminative, respectively. The experimental results of image transformation visualization, binary-way classification and face verification demonstrate that, by performing supervised learning, our models can greatly improve the performance. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. It is known that the inconsistent distribution and representation of different modalities, such as image and text, cause the heterogeneity gap that makes it challenging to correlate such heterogeneous data. Generative adversarial networks (GANs) have shown its strong ability of modeling data distribution and learning discriminative representation, existing GANs-based works mainly focus on generative problem to generate new data. We have different goal, aim to correlate heterogeneous data, by utilizing the power of GANs to model cross-modal joint distribution. Thus, we propose Cross-modal GANs to learn discriminative common representation for bridging heterogeneity gap. The main contributions are: (1) Cross-modal GANs architecture is proposed to model joint distribution over data of different modalities. The inter-modality and intra-modality correlation can be explored simultaneously in generative and discriminative models. Both of them beat each other to promote cross-modal correlation learning. (2) Cross-modal convolutional autoencoders with weight-sharing constraint are proposed to form generative model. They can not only exploit cross-modal correlation for learning common representation, but also preserve reconstruction information for capturing semantic consistency within each modality. (3) Cross-modal adversarial mechanism is proposed, which utilizes two kinds of discriminative models to simultaneously conduct intra-modality and inter-modality discrimination. They can mutually boost to make common representation more discriminative by adversarial training process. To the best of our knowledge, our proposed CM-GANs approach is the first to utilize GANs to perform cross-modal common representation learning. Experiments are conducted to verify the performance of our proposed approach on cross-modal retrieval paradigm, compared with 10 methods on 3 cross-modal datasets.