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Register a new userOnline matrix factorization for Markovian data and applications to Network Dictionary Learning
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Hanbaek Lyu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Online Matrix Factorization (OMF) is a fundamental tool for dictionary learning problems, giving an approximate representation of complex data sets in terms of a reduced number of extracted features. Convergence guarantees for most of the OMF algorithms in the literature assume independence between data matrices, and the case of dependent data streams remains largely unexplored. In this paper, we show that a non-convex generalization of the well-known OMF algorithm for i.i.d. stream of data in citep{mairal2010online} converges almost surely to the set of critical points of the expected loss function, even when the data matrices are functions of some underlying Markov chain satisfying a mild mixing condition. This allows one to extract features more efficiently from dependent data streams, as there is no need to subsample the data sequence to approximately satisfy the independence assumption. As the main application, by combining online non-negative matrix factorization and a recent MCMC algorithm for sampling motifs from networks, we propose a novel framework of Network Dictionary Learning, which extracts ``network dictionary patches from a given network in an online manner that encodes main features of the network. We demonstrate this technique and its application to network denoising problems on real-world network data.
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Online Tensor Factorization (OTF) is a fundamental tool in learning low-dimensional interpretable features from streaming multi-modal data. While various algorithmic and theoretical aspects of OTF have been investigated recently, general convergence guarantee to stationary points of the objective function without any incoherence or sparsity assumptions is still lacking even for the i.i.d. case. In this work, we introduce a novel OTF algorithm that learns a CANDECOMP/PARAFAC (CP) basis from a given stream of tensor-valued data under general constraints, including nonnegativity constraints that induce interpretability of learned CP basis. We prove that our algorithm converges almost surely to the set of stationary points of the objective function under the hypothesis that the sequence of data tensors is generated by some underlying Markov chain. Our setting covers the classical i.i.d. case as well as a wide range of application contexts including data streams generated by independent or MCMC sampling. Our result closes a gap between OTF and Online Matrix Factorization in global convergence analysis. Experimentally, we show that our OTF algorithm converges much faster than standard algorithms for nonnegative tensor factorization tasks on both synthetic and real-world data. Also, we demonstrate the utility of our algorithm on a diverse set of examples from image, video, and time-series data, illustrating how one may learn qualitatively different CP-dictionaries from the same tensor data by exploiting the tensor structure in multiple ways.
Online nonnegative matrix factorization (ONMF) is a matrix factorization technique in the online setting where data are acquired in a streaming fashion and the matrix factors are updated each time. This enables factor analysis to be performed concurrently with the arrival of new data samples. In this article, we demonstrate how one can use online nonnegative matrix factorization algorithms to learn joint dictionary atoms from an ensemble of correlated data sets. We propose a temporal dictionary learning scheme for time-series data sets, based on ONMF algorithms. We demonstrate our dictionary learning technique in the application contexts of historical temperature data, video frames, and color images.
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with $n$-dimensional orthonormal components in the presence of error with constant spectral norm (when viewed as an $n^2$-by-$n^2$ matrix). The running time is $n^5$ which is close to linear in the input size $n^4$. We also obtain algorithms with similar running time to learn sparsely-used orthogonal dictionaries even when feature representations have constant relative sparsity and non-independent coordinates. The only previous polynomial-time algorithms to solve these problem are based on solving large semidefinite programs. In contrast, our algorithms are easy to implement directly and are based on spectral projections and tensor-mode rearrangements. Or work is inspired by recent of Hopkins, Schramm, Shi, and Steurer (STOC16) that shows how fast spectral algorithms can achieve the guarantees of SOS for average-case problems. In this work, we introduce general techniques to capture the guarantees of SOS for worst-case problems.
Dictionary learning is a widely used unsupervised learning method in signal processing and machine learning. Most existing works of dictionary learning are in an offline manner. There are mainly two offline ways for dictionary learning. One is to do an alternative optimization of both the dictionary and the sparse code; the other way is to optimize the dictionary by restricting it over the orthogonal group. The latter one is called orthogonal dictionary learning which has a lower complexity implementation, hence, it is more favorable for lowcost devices. However, existing schemes on orthogonal dictionary learning only work with batch data and can not be implemented online, which is not applicable for real-time applications. This paper proposes a novel online orthogonal dictionary scheme to dynamically learn the dictionary from streaming data without storing the historical data. The proposed scheme includes a novel problem formulation and an efficient online algorithm design with convergence analysis. In the problem formulation, we relax the orthogonal constraint to enable an efficient online algorithm. In the algorithm design, we propose a new Frank-Wolfe-based online algorithm with a convergence rate of O(ln t/t^(1/4)). The convergence rate in terms of key system parameters is also derived. Experiments with synthetic data and real-world sensor readings demonstrate the effectiveness and efficiency of the proposed online orthogonal dictionary learning scheme.
Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability. We show that the resulting online algorithms have $O(sqrt{T})$ (approximate) regret under the full information setting. We further introduce a bandit extension of Blackwell approachability that we call Bandit Blackwell approachability. We leverage this notion to transform greedy robust offline algorithms into a $O(T^{2/3})$ (approximate) regret in the bandit setting. Demonstrating the flexibility of our framework, we apply our offline-to-online transformation to several problems at the intersection of revenue management, market design, and online optimization, including product ranking optimization in online platforms, reserve price optimization in auctions, and submodular maximization. We show that our transformation, when applied to these applications, leads to new regret bounds or improves the current known bounds.
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