5,338 research outputs found
Sparse Principal Component Analysis via Rotation and Truncation
Sparse principal component analysis (sparse PCA) aims at finding a sparse
basis to improve the interpretability over the dense basis of PCA, meanwhile
the sparse basis should cover the data subspace as much as possible. In
contrast to most of existing work which deal with the problem by adding some
sparsity penalties on various objectives of PCA, in this paper, we propose a
new method SPCArt, whose motivation is to find a rotation matrix and a sparse
basis such that the sparse basis approximates the basis of PCA after the
rotation. The algorithm of SPCArt consists of three alternating steps: rotate
PCA basis, truncate small entries, and update the rotation matrix. Its
performance bounds are also given. SPCArt is efficient, with each iteration
scaling linearly with the data dimension. It is easy to choose parameters in
SPCArt, due to its explicit physical explanations. Besides, we give a unified
view to several existing sparse PCA methods and discuss the connection with
SPCArt. Some ideas in SPCArt are extended to GPower, a popular sparse PCA
algorithm, to overcome its drawback. Experimental results demonstrate that
SPCArt achieves the state-of-the-art performance. It also achieves a good
tradeoff among various criteria, including sparsity, explained variance,
orthogonality, balance of sparsity among loadings, and computational speed
On the Worst-Case Approximability of Sparse PCA
It is well known that Sparse PCA (Sparse Principal Component Analysis) is
NP-hard to solve exactly on worst-case instances. What is the complexity of
solving Sparse PCA approximately? Our contributions include: 1) a simple and
efficient algorithm that achieves an -approximation; 2) NP-hardness
of approximation to within , for some small constant
; 3) SSE-hardness of approximation to within any constant
factor; and 4) an
("quasi-quasi-polynomial") gap for the standard semidefinite program.Comment: 20 page
Sparse eigenbasis approximation: multiple feature extraction across spatiotemporal scales with application to coherent set identification
The output of spectral clustering is a collection of eigenvalues and
eigenvectors that encode important connectivity information about a graph or a
manifold. This connectivity information is often not cleanly represented in the
eigenvectors and must be disentangled by some secondary procedure. We propose
the use of an approximate sparse basis for the space spanned by the leading
eigenvectors as a natural, robust, and efficient means of performing this
separation. The use of sparsity yields a natural cutoff in this disentanglement
procedure and is particularly useful in practical situations when there is no
clear eigengap. In order to select a suitable collection of vectors we develop
a new Weyl-inspired eigengap heuristic and heuristics based on the sparse basis
vectors. We develop an automated eigenvector separation procedure and
illustrate its efficacy on examples from time-dependent dynamics on manifolds.
In this context, transfer operator approaches are extensively used to find
dynamically disconnected regions of phase space, known as almost-invariant sets
or coherent sets. The dominant eigenvectors of transfer operators or related
operators, such as the dynamic Laplacian, encode dynamic connectivity
information. Our sparse eigenbasis approximation (SEBA) methodology streamlines
the final stage of transfer operator methods, namely the extraction of
almost-invariant or coherent sets from the eigenvectors. It is particularly
useful when used on domains with large numbers of coherent sets, and when the
coherent sets do not exhaust the phase space, such as in large geophysical
datasets
Spectral Sparse Representation for Clustering: Evolved from PCA, K-means, Laplacian Eigenmap, and Ratio Cut
Dimensionality reduction, cluster analysis, and sparse representation are
basic components in machine learning. However, their relationships have not yet
been fully investigated. In this paper, we find that the spectral graph theory
underlies a series of these elementary methods and can unify them into a
complete framework. The methods include PCA, K-means, Laplacian eigenmap (LE),
ratio cut (Rcut), and a new sparse representation method developed by us,
called spectral sparse representation (SSR). Further, extended relations to
conventional over-complete sparse representations (e.g., method of optimal
directions, KSVD), manifold learning (e.g., kernel PCA, multidimensional
scaling, Isomap, locally linear embedding), and subspace clustering (e.g.,
sparse subspace clustering, low-rank representation) are incorporated. We show
that, under an ideal condition from the spectral graph theory, PCA, K-means,
LE, and Rcut are unified together. And when the condition is relaxed, the
unification evolves to SSR, which lies in the intermediate between PCA/LE and
K-mean/Rcut. An efficient algorithm, NSCrt, is developed to solve the sparse
codes of SSR. SSR combines merits of both sides: its sparse codes reduce
dimensionality of data meanwhile revealing cluster structure. For its inherent
relation to cluster analysis, the codes of SSR can be directly used for
clustering. Scut, a clustering approach derived from SSR reaches the
state-of-the-art performance in the spectral clustering family. The one-shot
solution obtained by Scut is comparable to the optimal result of K-means that
are run many times. Experiments on various data sets demonstrate the properties
and strengths of SSR, NSCrt, and Scut
A Fast deflation Method for Sparse Principal Component Analysis via Subspace Projections
The implementation of conventional sparse principal component analysis (SPCA)
on high-dimensional data sets has become a time consuming work. In this paper,
a series of subspace projections are constructed efficiently by using Household
QR factorization. With the aid of these subspace projections, a fast deflation
method, called SPCA-SP, is developed for SPCA. This method keeps a good
tradeoff between various criteria, including sparsity, orthogonality, explained
variance, balance of sparsity, and computational cost. Comparative experiments
on the benchmark data sets confirm the effectiveness of the proposed method.Comment: 4 figures, 2 table
Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview
Substantial progress has been made recently on developing provably accurate
and efficient algorithms for low-rank matrix factorization via nonconvex
optimization. While conventional wisdom often takes a dim view of nonconvex
optimization algorithms due to their susceptibility to spurious local minima,
simple iterative methods such as gradient descent have been remarkably
successful in practice. The theoretical footings, however, had been largely
lacking until recently.
In this tutorial-style overview, we highlight the important role of
statistical models in enabling efficient nonconvex optimization with
performance guarantees. We review two contrasting approaches: (1) two-stage
algorithms, which consist of a tailored initialization step followed by
successive refinement; and (2) global landscape analysis and
initialization-free algorithms. Several canonical matrix factorization problems
are discussed, including but not limited to matrix sensing, phase retrieval,
matrix completion, blind deconvolution, robust principal component analysis,
phase synchronization, and joint alignment. Special care is taken to illustrate
the key technical insights underlying their analyses. This article serves as a
testament that the integrated consideration of optimization and statistics
leads to fruitful research findings.Comment: Invited overview articl
Why (and How) Avoid Orthogonal Procrustes in Regularized Multivariate Analysis
Multivariate Analysis (MVA) comprises a family of well-known methods for
feature extraction that exploit correlations among input variables of the data
representation. One important property that is enjoyed by most such methods is
uncorrelation among the extracted features. Recently, regularized versions of
MVA methods have appeared in the literature, mainly with the goal to gain
interpretability of the solution. In these cases, the solutions can no longer
be obtained in a closed manner, and it is frequent to recur to the iteration of
two steps, one of them being an orthogonal Procrustes problem. This letter
shows that the Procrustes solution is not optimal from the perspective of the
overall MVA method, and proposes an alternative approach based on the solution
of an eigenvalue problem. Our method ensures the preservation of several
properties of the original methods, most notably the uncorrelation of the
extracted features, as demonstrated theoretically and through a collection of
selected experiments.Comment: 9 pages; added acknowledgment
Optimal linear estimation under unknown nonlinear transform
Linear regression studies the problem of estimating a model parameter
, from observations
from linear model . We consider a significant
generalization in which the relationship between and is noisy, quantized to a single bit, potentially nonlinear,
noninvertible, as well as unknown. This model is known as the single-index
model in statistics, and, among other things, it represents a significant
generalization of one-bit compressed sensing. We propose a novel spectral-based
estimation procedure and show that we can recover in settings (i.e.,
classes of link function ) where previous algorithms fail. In general, our
algorithm requires only very mild restrictions on the (unknown) functional
relationship between and . We also
consider the high dimensional setting where is sparse ,and introduce
a two-stage nonconvex framework that addresses estimation challenges in high
dimensional regimes where . For a broad class of link functions
between and , we establish minimax
lower bounds that demonstrate the optimality of our estimators in both the
classical and high dimensional regimes.Comment: 25 pages, 3 figure
Implementing smooth functions of a Hermitian matrix on a quantum computer
We review existing methods for implementing smooth functions f(A) of a sparse
Hermitian matrix A on a quantum computer, and analyse a further combination of
these techniques which has some advantages of simplicity and resource
consumption in some cases. Our construction uses the linear combination of
unitaries method with Chebyshev polynomial approximations. The query complexity
we obtain is O(log C/eps) where eps is the approximation precision, and C>0 is
an upper bound on the magnitudes of the derivatives of the function f over the
domain of interest. The success probability depends on the 1-norm of the Taylor
series coefficients of f, the sparsity d of the matrix, and inversely on the
smallest singular value of the target matrix f(A).Comment: 16 page
Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1 -- Philosophy, Features and Application for (Parametric) Model
Matrix equations are omnipresent in (numerical) linear algebra and systems
theory. Especially in model order reduction (MOR) they play a key role in many
balancing based reduction methods for linear dynamical systems. When these
systems arise from spatial discretizations of evolutionary partial differential
equations, their coefficient matrices are typically large and sparse. Moreover,
the numbers of inputs and outputs of these systems are typically far smaller
than the number of spatial degrees of freedom. Then, in many situations the
solutions of the corresponding large-scale matrix equations are observed to
have low (numerical) rank. This feature is exploited by M-M.E.S.S. to find
successively larger low-rank factorizations approximating the solutions. This
contribution describes the basic philosophy behind the implementation and the
features of the package, as well as its application in the model order
reduction of large-scale linear time-invariant (LTI) systems and parametric LTI
systems.Comment: 18 pages, 4 figures, 5 table
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