36 research outputs found
CUR Algorithm with Incomplete Matrix Observation
CUR matrix decomposition is a randomized algorithm that can efficiently
compute the low rank approximation for a given rectangle matrix. One limitation
with the existing CUR algorithms is that they require an access to the full
matrix A for computing U. In this work, we aim to alleviate this limitation. In
particular, we assume that besides having an access to randomly sampled d rows
and d columns from A, we only observe a subset of randomly sampled entries from
A. Our goal is to develop a low rank approximation algorithm, similar to CUR,
based on (i) randomly sampled rows and columns from A, and (ii) randomly
sampled entries from A. The proposed algorithm is able to perfectly recover the
target matrix A with only O(rn log n) number of observed entries. In addition,
instead of having to solve an optimization problem involved trace norm
regularization, the proposed algorithm only needs to solve a standard
regression problem. Finally, unlike most matrix completion theories that hold
only when the target matrix is of low rank, we show a strong guarantee for the
proposed algorithm even when the target matrix is not low rank
Scaled Nuclear Norm Minimization for Low-Rank Tensor Completion
Minimizing the nuclear norm of a matrix has been shown to be very efficient
in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of
nuclear norms of matricizations of a tensor has been shown to be very efficient
in recovering a low-Tucker-rank sampled tensor. In this paper, we propose to
recover a low-TT-rank sampled tensor by minimizing a weighted sum of nuclear
norms of unfoldings of the tensor. We provide numerical results to show that
our proposed method requires significantly less number of samples to recover to
the original tensor in comparison with simply minimizing the sum of nuclear
norms since the structure of the unfoldings in the TT tensor model is
fundamentally different from that of matricizations in the Tucker tensor model
Subspace Learning from Extremely Compressed Measurements
We consider learning the principal subspace of a large set of vectors from an
extremely small number of compressive measurements of each vector. Our
theoretical results show that even a constant number of measurements per column
suffices to approximate the principal subspace to arbitrary precision, provided
that the number of vectors is large. This result is achieved by a simple
algorithm that computes the eigenvectors of an estimate of the covariance
matrix. The main insight is to exploit an averaging effect that arises from
applying a different random projection to each vector. We provide a number of
simulations confirming our theoretical results
Matrix Completion with Sparse Noisy Rows
Exact matrix completion and low rank matrix estimation problems has been
studied in different underlying conditions. In this work we study exact
low-rank completion under non-degenerate noise model. Non-degenerate random
noise model has been previously studied by many researchers under given
condition that the noise is sparse and existing in some of the columns. In this
paper, we assume that each row can receive random noise instead of columns and
propose an interactive algorithm that is robust to this noise. We show that we
use a parametrization technique to give a condition when the underlying matrix
could be recoverable and suggest an algorithm which recovers the underlying
matrix
Tensor Matched Kronecker-Structured Subspace Detection for Missing Information
We consider the problem of detecting whether a tensor signal having many
missing entities lies within a given low dimensional Kronecker-Structured (KS)
subspace. This is a matched subspace detection problem. Tensor matched subspace
detection problem is more challenging because of the intertwined signal
dimensions. We solve this problem by projecting the signal onto the Kronecker
structured subspace, which is a Kronecker product of different subspaces
corresponding to each signal dimension. Under this framework, we define the KS
subspaces and the orthogonal projection of the signal onto the KS subspace. We
prove that reliable detection is possible as long as the cardinality of the
missing signal is greater than the dimensions of the KS subspace by bounding
the residual energy of the sampling signal with high probability
Matrix Completion from Non-Uniformly Sampled Entries
In this paper, we consider matrix completion from non-uniformly sampled
entries including fully observed and partially observed columns. Specifically,
we assume that a small number of columns are randomly selected and fully
observed, and each remaining column is partially observed with uniform
sampling. To recover the unknown matrix, we first recover its column space from
the fully observed columns. Then, for each partially observed column, we
recover it by finding a vector which lies in the recovered column space and
consists of the observed entries. When the unknown matrix is
low-rank, we show that our algorithm can exactly recover it from merely
entries, where is the rank of the matrix. Furthermore,
for a noisy low-rank matrix, our algorithm computes a low-rank approximation of
the unknown matrix and enjoys an additive error bound measured by Frobenius
norm. Experimental results on synthetic datasets verify our theoretical claims
and demonstrate the effectiveness of our proposed algorithm
An algorithm for online tensor prediction
We present a new method for online prediction and learning of tensors
(-way arrays, ) from sequential measurements. We focus on the specific
case of 3-D tensors and exploit a recently developed framework of structured
tensor decompositions proposed in [1]. In this framework it is possible to
treat 3-D tensors as linear operators and appropriately generalize notions of
rank and positive definiteness to tensors in a natural way. Using these notions
we propose a generalization of the matrix exponentiated gradient descent
algorithm [2] to a tensor exponentiated gradient descent algorithm using an
extension of the notion of von-Neumann divergence to tensors. Then following a
similar construction as in [3], we exploit this algorithm to propose an online
algorithm for learning and prediction of tensors with provable regret
guarantees. Simulations results are presented on semi-synthetic data sets of
ratings evolving in time under local influence over a social network. The
result indicate superior performance compared to other (online) convex tensor
completion methods
Tensor Matched Subspace Detection
The problem of testing whether a signal lies within a given subspace, also
named matched subspace detection, has been well studied when the signal is
represented as a vector. However, the matched subspace detection methods based
on vectors can not be applied to the situations that signals are naturally
represented as multi-dimensional data arrays or tensors. Considering that
tensor subspaces and orthogonal projections onto these subspaces are well
defined in the recently proposed transform-based tensor model, which motivates
us to investigate the problem of matched subspace detection in high dimensional
case. In this paper, we propose an approach for tensor matched subspace
detection based on the transform-based tensor model with tubal-sampling and
elementwise-sampling, respectively. First, we construct estimators based on
tubal-sampling and elementwise-sampling to estimate the energy of a signal
outside a given subspace of a third-order tensor and then give the probability
bounds of our estimators, which show that our estimators work effectively when
the sample size is greater than a constant. Secondly, the detectors both for
noiseless data and noisy data are given, and the corresponding detection
performance analyses are also provided. Finally, based on discrete Fourier
transform (DFT) and discrete cosine transform (DCT), the performance of our
estimators and detectors are evaluated by several simulations, and simulation
results verify the effectiveness of our approach
Compact Factorization of Matrices Using Generalized Round-Rank
Matrix factorization is a well-studied task in machine learning for compactly
representing large, noisy data. In our approach, instead of using the
traditional concept of matrix rank, we define a new notion of link-rank based
on a non-linear link function used within factorization. In particular, by
applying the round function on a factorization to obtain ordinal-valued
matrices, we introduce generalized round-rank (GRR). We show that not only are
there many full-rank matrices that are low GRR, but further, that these
matrices cannot be approximated well by low-rank linear factorization. We
provide uniqueness conditions of this formulation and provide gradient
descent-based algorithms. Finally, we present experiments on real-world
datasets to demonstrate that the GRR-based factorization is significantly more
accurate than linear factorization, while converging faster and using lower
rank representations
Relaxed Leverage Sampling for Low-rank Matrix Completion
We consider the problem of exact recovery of any matrix of rank
from a small number of observed entries via the standard nuclear norm
minimization framework. Such low-rank matrices have degrees of freedom
. We show that any arbitrary low-rank matrices can be
recovered exactly from a randomly sampled entries, thus matching the lower
bound on the required number of entries (in terms of degrees of freedom), with
an additional factor of . To achieve this bound on sample size
we observe each entry with probabilities proportional to the sum of
corresponding row and column leverage scores, minus their product. We show that
this relaxation in sampling probabilities (as opposed to sum of leverage scores
in Chen et al, 2014) can give us an additive
improvement on the (best known) sample size obtained by Chen et al, 2014, for
the nuclear norm minimization. Experiments on real data corroborate the
theoretical improvement on sample size. Further, exact recovery of
incoherent matrices (with restricted leverage scores), and matrices with
only one of the row or column spaces to be incoherent, can be performed using
our relaxed leverage score sampling, via nuclear norm minimization, without
knowing the leverage scores a priori. In such settings also we can achieve
improvement on sample size