343 research outputs found
Matrix Recipes for Hard Thresholding Methods
In this paper, we present and analyze a new set of low-rank recovery
algorithms for linear inverse problems within the class of hard thresholding
methods. We provide strategies on how to set up these algorithms via basic
ingredients for different configurations to achieve complexity vs. accuracy
tradeoffs. Moreover, we study acceleration schemes via memory-based techniques
and randomized, -approximate matrix projections to decrease the
computational costs in the recovery process. For most of the configurations, we
present theoretical analysis that guarantees convergence under mild problem
conditions. Simulation results demonstrate notable performance improvements as
compared to state-of-the-art algorithms both in terms of reconstruction
accuracy and computational complexity.Comment: 26 page
Robust Linear Regression Analysis - A Greedy Approach
The task of robust linear estimation in the presence of outliers is of
particular importance in signal processing, statistics and machine learning.
Although the problem has been stated a few decades ago and solved using
classical (considered nowadays) methods, recently it has attracted more
attention in the context of sparse modeling, where several notable
contributions have been made. In the present manuscript, a new approach is
considered in the framework of greedy algorithms. The noise is split into two
components: a) the inlier bounded noise and b) the outliers, which are
explicitly modeled by employing sparsity arguments. Based on this scheme, a
novel efficient algorithm (Greedy Algorithm for Robust Denoising - GARD), is
derived. GARD alternates between a least square optimization criterion and an
Orthogonal Matching Pursuit (OMP) selection step that identifies the outliers.
The case where only outliers are present has been studied separately, where
bounds on the \textit{Restricted Isometry Property} guarantee that the recovery
of the signal via GARD is exact. Moreover, theoretical results concerning
convergence as well as the derivation of error bounds in the case of additional
bounded noise are discussed. Finally, we provide extensive simulations, which
demonstrate the comparative advantages of the new technique
CUR Decompositions, Similarity Matrices, and Subspace Clustering
A general framework for solving the subspace clustering problem using the CUR
decomposition is presented. The CUR decomposition provides a natural way to
construct similarity matrices for data that come from a union of unknown
subspaces . The similarity
matrices thus constructed give the exact clustering in the noise-free case.
Additionally, this decomposition gives rise to many distinct similarity
matrices from a given set of data, which allow enough flexibility to perform
accurate clustering of noisy data. We also show that two known methods for
subspace clustering can be derived from the CUR decomposition. An algorithm
based on the theoretical construction of similarity matrices is presented, and
experiments on synthetic and real data are presented to test the method.
Additionally, an adaptation of our CUR based similarity matrices is utilized
to provide a heuristic algorithm for subspace clustering; this algorithm yields
the best overall performance to date for clustering the Hopkins155 motion
segmentation dataset.Comment: Approximately 30 pages. Current version contains improved algorithm
and numerical experiments from the previous versio
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
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