arXiv subjects

Rong Ge

Publications and source records attributed to Rong Ge.

99 records · Page 6Linked to original sources

arXiv2014

More Algorithms for Provable Dictionary Learning

In dictionary learning, also known as sparse coding, the algorithm is given samples of the form $y = Ax$ where $x\in \mathbb{R}^m$ is an unknown random sparse vector and $A$ is an unknown dictionary matrix in $\mathbb{R}^{n\times m}$ (usually $m > n$, which is the overcomplete case). The goal is to learn $A$ and $x$. This problem has been studied in neuroscience, machine learning, visions, and image processing. In practice it is solved by heuristic algorithms and provable algorithms seemed hard to find. Recently, provable algorithms were found that work if the unknown feature vector $x$ is $\sqrt{n}$-sparse or even sparser. Spielman et al. \cite{DBLP:journals/jmlr/SpielmanWW12} did this for dictionaries where $m=n$; Arora et al. \cite{AGM} gave an algorithm for overcomplete ($m >n$) and incoherent matrices $A$; and Agarwal et al. \cite{DBLP:journals/corr/AgarwalAN13} handled a similar case but with weaker guarantees. This raised the problem of designing provable algorithms that allow sparsity $\gg \sqrt{n}$ in the hidden vector $x$. The current paper designs algorithms that allow sparsity up to $n/poly(\log n)$. It works for a class of matrices where features are individually recoverable, a new notion identified in this paper that may motivate further work. The algorithm runs in quasipolynomial time because they use limited enumeration.

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arXiv2013

A Tensor Approach to Learning Mixed Membership Community Models

Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed community detection for a family of probabilistic network models with overlapping communities, termed as the mixed membership Dirichlet model, first introduced by Airoldi et al. This model allows for nodes to have fractional memberships in multiple communities and assumes that the community memberships are drawn from a Dirichlet distribution. Moreover, it contains the stochastic block model as a special case. We propose a unified approach to learning these models via a tensor spectral decomposition method. Our estimator is based on low-order moment tensor of the observed network, consisting of 3-star counts. Our learning method is fast and is based on simple linear algebraic operations, e.g. singular value decomposition and tensor power iterations. We provide guaranteed recovery of community memberships and model parameters and present a careful finite sample analysis of our learning method. As an important special case, our results match the best known scaling requirements for the (homogeneous) stochastic block model.

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arXiv2013

Provable Bounds for Learning Some Deep Representations

We give algorithms with provable guarantees that learn a class of deep nets in the generative model view popularized by Hinton and others. Our generative model is an $n$ node multilayer neural net that has degree at most $n^γ$ for some $γ<1$ and each edge has a random edge weight in $[-1,1]$. Our algorithm learns {\em almost all} networks in this class with polynomial running time. The sample complexity is quadratic or cubic depending upon the details of the model. The algorithm uses layerwise learning. It is based upon a novel idea of observing correlations among features and using these to infer the underlying edge structure via a global graph recovery procedure. The analysis of the algorithm reveals interesting structure of neural networks with random edge weights.

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arXiv2013

Towards a better approximation for sparsest cut?

We give a new $(1+ε)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+ε)$-approximation on such general family of graphs.

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arXiv2012

A Practical Algorithm for Topic Modeling with Provable Guarantees

Topic models provide a useful method for dimensionality reduction and exploratory data analysis in large text corpora. Most approaches to topic model inference have been based on a maximum likelihood objective. Efficient algorithms exist that approximate this objective, but they have no provable guarantees. Recently, algorithms have been introduced that provide provable bounds, but these algorithms are not practical because they are inefficient and not robust to violations of model assumptions. In this paper we present an algorithm for topic model inference that is both provable and practical. The algorithm produces results comparable to the best MCMC implementations while running orders of magnitude faster.

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arXiv2012

Provable ICA with Unknown Gaussian Noise, and Implications for Gaussian Mixtures and Autoencoders

We present a new algorithm for Independent Component Analysis (ICA) which has provable performance guarantees. In particular, suppose we are given samples of the form $y = Ax + η$ where $A$ is an unknown $n \times n$ matrix and $x$ is a random variable whose components are independent and have a fourth moment strictly less than that of a standard Gaussian random variable and $η$ is an $n$-dimensional Gaussian random variable with unknown covariance $Σ$: We give an algorithm that provable recovers $A$ and $Σ$ up to an additive $ε$ and whose running time and sample complexity are polynomial in $n$ and $1 / ε$. To accomplish this, we introduce a novel "quasi-whitening" step that may be useful in other contexts in which the covariance of Gaussian noise is not known in advance. We also give a general framework for finding all local optima of a function (given an oracle for approximately finding just one) and this is a crucial step in our algorithm, one that has been overlooked in previous attempts, and allows us to control the accumulation of error when we find the columns of $A$ one by one via local search.

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arXiv2012

Learning Topic Models - Going beyond SVD

Topic Modeling is an approach used for automatic comprehension and classification of data in a variety of settings, and perhaps the canonical application is in uncovering thematic structure in a corpus of documents. A number of foundational works both in machine learning and in theory have suggested a probabilistic model for documents, whereby documents arise as a convex combination of (i.e. distribution on) a small number of topic vectors, each topic vector being a distribution on words (i.e. a vector of word-frequencies). Similar models have since been used in a variety of application areas; the Latent Dirichlet Allocation or LDA model of Blei et al. is especially popular. Theoretical studies of topic modeling focus on learning the model's parameters assuming the data is actually generated from it. Existing approaches for the most part rely on Singular Value Decomposition(SVD), and consequently have one of two limitations: these works need to either assume that each document contains only one topic, or else can only recover the span of the topic vectors instead of the topic vectors themselves. This paper formally justifies Nonnegative Matrix Factorization(NMF) as a main tool in this context, which is an analog of SVD where all vectors are nonnegative. Using this tool we give the first polynomial-time algorithm for learning topic models without the above two limitations. The algorithm uses a fairly mild assumption about the underlying topic matrix called separability, which is usually found to hold in real-life data. A compelling feature of our algorithm is that it generalizes to models that incorporate topic-topic correlations, such as the Correlated Topic Model and the Pachinko Allocation Model. We hope that this paper will motivate further theoretical results that use NMF as a replacement for SVD - just as NMF has come to replace SVD in many applications.

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arXiv2011

Finding Overlapping Communities in Social Networks: Toward a Rigorous Approach

A "community" in a social network is usually understood to be a group of nodes more densely connected with each other than with the rest of the network. This is an important concept in most domains where networks arise: social, technological, biological, etc. For many years algorithms for finding communities implicitly assumed communities are nonoverlapping (leading to use of clustering-based approaches) but there is increasing interest in finding overlapping communities. A barrier to finding communities is that the solution concept is often defined in terms of an NP-complete problem such as Clique or Hierarchical Clustering. This paper seeks to initiate a rigorous approach to the problem of finding overlapping communities, where "rigorous" means that we clearly state the following: (a) the object sought by our algorithm (b) the assumptions about the underlying network (c) the (worst-case) running time. Our assumptions about the network lie between worst-case and average-case. An average case analysis would require a precise probabilistic model of the network, on which there is currently no consensus. However, some plausible assumptions about network parameters can be gleaned from a long body of work in the sociology community spanning five decades focusing on the study of individual communities and ego-centric networks. Thus our assumptions are somewhat "local" in nature. Nevertheless they suffice to permit a rigorous analysis of running time of algorithms that recover global structure. Our algorithms use random sampling similar to that in property testing and algorithms for dense graphs. However, our networks are not necessarily dense graphs, not even in local neighborhoods. Our algorithms explore a local-global relationship between ego-centric and socio-centric networks that we hope will provide a fruitful framework for future work both in computer science and sociology.

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arXiv2011

Computing a Nonnegative Matrix Factorization -- Provably

In the Nonnegative Matrix Factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$ where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r \times m$ respectively. In some applications, it makes sense to ask instead for the product $AW$ to approximate $M$ -- i.e. (approximately) minimize $\norm{M - AW}_F$ where $\norm{}_F$ denotes the Frobenius norm; we refer to this as Approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where $A$ and $W$ are computed using a variety of local search heuristics. Vavasis proved that this problem is NP-complete. We initiate a study of when this problem is solvable in polynomial time: 1. We give a polynomial-time algorithm for exact and approximate NMF for every constant $r$. Indeed NMF is most interesting in applications precisely when $r$ is small. 2. We complement this with a hardness result, that if exact NMF can be solved in time $(nm)^{o(r)}$, 3-SAT has a sub-exponential time algorithm. This rules out substantial improvements to the above algorithm. 3. We give an algorithm that runs in time polynomial in $n$, $m$ and $r$ under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesting and important direction for future work.

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