190 research outputs found
Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
The proposed article aims at offering a comprehensive tutorial for the
computational aspects of structured matrix and tensor factorization. Unlike
existing tutorials that mainly focus on {\it algorithmic procedures} for a
small set of problems, e.g., nonnegativity or sparsity-constrained
factorization, we take a {\it top-down} approach: we start with general
optimization theory (e.g., inexact and accelerated block coordinate descent,
stochastic optimization, and Gauss-Newton methods) that covers a wide range of
factorization problems with diverse constraints and regularization terms of
engineering interest. Then, we go `under the hood' to showcase specific
algorithm design under these introduced principles. We pay a particular
attention to recent algorithmic developments in structured tensor and matrix
factorization (e.g., random sketching and adaptive step size based stochastic
optimization and structure-exploiting second-order algorithms), which are the
state of the art---yet much less touched upon in the literature compared to
{\it block coordinate descent} (BCD)-based methods. We expect that the article
to have an educational values in the field of structured factorization and hope
to stimulate more research in this important and exciting direction.Comment: Final Version; to appear in IEEE Signal Processing Magazine; title
revised to comply with the journal's rul
New Approaches in Multi-View Clustering
Many real-world datasets can be naturally described by multiple views. Due to this, multi-view learning has drawn much attention from both academia and industry. Compared to single-view learning, multi-view learning has demonstrated plenty of advantages. Clustering has long been serving as a critical technique in data mining and machine learning. Recently, multi-view clustering has achieved great success in various applications. To provide a comprehensive review of the typical multi-view clustering methods and their corresponding recent developments, this chapter summarizes five kinds of popular clustering methods and their multi-view learning versions, which include k-means, spectral clustering, matrix factorization, tensor decomposition, and deep learning. These clustering methods are the most widely employed algorithms for single-view data, and lots of efforts have been devoted to extending them for multi-view clustering. Besides, many other multi-view clustering methods can be unified into the frameworks of these five methods. To promote further research and development of multi-view clustering, some popular and open datasets are summarized in two categories. Furthermore, several open issues that deserve more exploration are pointed out in the end
Characterizing Variability of Modular Brain Connectivity with Constrained Principal Component Analysis
Characterizing the variability of resting-state functional brain connectivity across subjects and/or over time has recently attracted much attention. Principal component analysis (PCA) serves as a fundamental statistical technique for such analyses. However, performing PCA on high-dimensional connectivity matrices yields complicated "eigenconnectivity" patterns, for which systematic interpretation is a challenging issue. Here, we overcome this issue with a novel constrained PCA method for connectivity matrices by extending the idea of the previously proposed orthogonal connectivity factorization method. Our new method, modular connectivity factorization (MCF), explicitly introduces the modularity of brain networks as a parametric constraint on eigenconnectivity matrices. In particular, MCF analyzes the variability in both intra-and inter-module connectivities, simultaneously finding network modules in a principled, data-driven manner. The parametric constraint provides a compact module based visualization scheme with which the result can be intuitively interpreted. We develop an optimization algorithm to solve the constrained PCA problem and validate our method in simulation studies and with a resting-state functional connectivity MRI dataset of 986 subjects. The results show that the proposed MCF method successfully reveals the underlying modular eigenconnectivity patterns in more general situations and is a promising alternative to existing methods.Peer reviewe
Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction
It is difficult to find the optimal sparse solution of a manifold learning
based dimensionality reduction algorithm. The lasso or the elastic net
penalized manifold learning based dimensionality reduction is not directly a
lasso penalized least square problem and thus the least angle regression (LARS)
(Efron et al. \cite{LARS}), one of the most popular algorithms in sparse
learning, cannot be applied. Therefore, most current approaches take indirect
ways or have strict settings, which can be inconvenient for applications. In
this paper, we proposed the manifold elastic net or MEN for short. MEN
incorporates the merits of both the manifold learning based dimensionality
reduction and the sparse learning based dimensionality reduction. By using a
series of equivalent transformations, we show MEN is equivalent to the lasso
penalized least square problem and thus LARS is adopted to obtain the optimal
sparse solution of MEN. In particular, MEN has the following advantages for
subsequent classification: 1) the local geometry of samples is well preserved
for low dimensional data representation, 2) both the margin maximization and
the classification error minimization are considered for sparse projection
calculation, 3) the projection matrix of MEN improves the parsimony in
computation, 4) the elastic net penalty reduces the over-fitting problem, and
5) the projection matrix of MEN can be interpreted psychologically and
physiologically. Experimental evidence on face recognition over various popular
datasets suggests that MEN is superior to top level dimensionality reduction
algorithms.Comment: 33 pages, 12 figure
Finding a low-rank basis in a matrix subspace
For a given matrix subspace, how can we find a basis that consists of
low-rank matrices? This is a generalization of the sparse vector problem. It
turns out that when the subspace is spanned by rank-1 matrices, the matrices
can be obtained by the tensor CP decomposition. For the higher rank case, the
situation is not as straightforward. In this work we present an algorithm based
on a greedy process applicable to higher rank problems. Our algorithm first
estimates the minimum rank by applying soft singular value thresholding to a
nuclear norm relaxation, and then computes a matrix with that rank using the
method of alternating projections. We provide local convergence results, and
compare our algorithm with several alternative approaches. Applications include
data compression beyond the classical truncated SVD, computing accurate
eigenvectors of a near-multiple eigenvalue, image separation and graph
Laplacian eigenproblems
Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols
In recent years, the polynomial method from circuit complexity has been applied to several fundamental problems and obtains the state-of-the-art running times (e.g., R. Williams\u27s n^3 / 2^{Omega(sqrt{log n})} time algorithm for APSP). As observed in [Alman and Williams, STOC 2017], almost all applications of the polynomial method in algorithm design ultimately rely on certain (probabilistic) low-rank decompositions of the computation matrices corresponding to key subroutines. They suggest that making use of low-rank decompositions directly could lead to more powerful algorithms, as the polynomial method is just one way to derive such a decomposition.
Inspired by their observation, in this paper, we study another way of systematically constructing low-rank decompositions of matrices which could be used by algorithms - communication protocols. Since their introduction, it is known that various types of communication protocols lead to certain low-rank decompositions (e.g., P protocols/rank, BQP protocols/approximate rank). These are usually interpreted as approaches for proving communication lower bounds, while in this work we explore the other direction.
We have the following two generic algorithmic applications of communication protocols:
- Quantum Communication Protocols and Deterministic Approximate Counting. Our first connection is that a fast BQP communication protocol for a function f implies a fast deterministic additive approximate counting algorithm for a related pair counting problem. Applying known BQP communication protocols, we get fast deterministic additive approximate counting algorithms for Count-OV (#OV), Sparse Count-OV and Formula of SYM circuits. In particular, our approximate counting algorithm for #OV runs in near-linear time for all dimensions d = o(log^2 n). Previously, even no truly-subquadratic time algorithm was known for d = omega(log n).
- Arthur-Merlin Communication Protocols and Faster Satisfying-Pair Algorithms. Our second connection is that a fast AM^{cc} protocol for a function f implies a faster-than-bruteforce algorithm for f-Satisfying-Pair. Using the classical Goldwasser-Sisper AM protocols for approximating set size, we obtain a new algorithm for approximate Max-IP_{n,c log n} in time n^{2 - 1/O(log c)}, matching the state-of-the-art algorithms in [Chen, CCC 2018].
We also apply our second connection to shed some light on long-standing open problems in communication complexity. We show that if the Longest Common Subsequence (LCS) problem admits a fast (computationally efficient) AM^{cc} protocol (polylog(n) complexity), then polynomial-size Formula-SAT admits a 2^{n - n^{1-delta}} time algorithm for any constant delta > 0, which is conjectured to be unlikely by a recent work [Abboud and Bringmann, ICALP 2018]. The same holds even for a fast (computationally efficient) PH^{cc} protocol
Privacy-Preserving Tensor Factorization for Collaborative Health Data Analysis
Tensor factorization has been demonstrated as an efficient approach for
computational phenotyping, where massive electronic health records (EHRs) are
converted to concise and meaningful clinical concepts. While distributing the
tensor factorization tasks to local sites can avoid direct data sharing, it
still requires the exchange of intermediary results which could reveal
sensitive patient information. Therefore, the challenge is how to jointly
decompose the tensor under rigorous and principled privacy constraints, while
still support the model's interpretability. We propose DPFact, a
privacy-preserving collaborative tensor factorization method for computational
phenotyping using EHR. It embeds advanced privacy-preserving mechanisms with
collaborative learning. Hospitals can keep their EHR database private but also
collaboratively learn meaningful clinical concepts by sharing differentially
private intermediary results. Moreover, DPFact solves the heterogeneous patient
population using a structured sparsity term. In our framework, each hospital
decomposes its local tensors, and sends the updated intermediary results with
output perturbation every several iterations to a semi-trusted server which
generates the phenotypes. The evaluation on both real-world and synthetic
datasets demonstrated that under strict privacy constraints, our method is more
accurate and communication-efficient than state-of-the-art baseline methods
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