153 research outputs found
A large deviation inequality for the rank of a random matrix
Let be an random matrix with independent identically
distributed non-constant subgaussian entries.Then for any ,
with probability at least
Small ball probability for the condition number of random matrices
Let be an random matrix with i.i.d. entries of zero mean,
unit variance and a bounded subgaussian moment. We show that the condition
number satisfies the small ball probability estimate
where may only depend on the subgaussian moment.
Although the estimate can be obtained as a combination of known results and
techniques, it was not noticed in the literature before. As a key step of the
proof, we apply estimates for the singular values of , obtained (under some additional assumptions) by Nguyen.Comment: Some changes according to the Referee's comment
Block CUR: Decomposing Matrices using Groups of Columns
A common problem in large-scale data analysis is to approximate a matrix
using a combination of specifically sampled rows and columns, known as CUR
decomposition. Unfortunately, in many real-world environments, the ability to
sample specific individual rows or columns of the matrix is limited by either
system constraints or cost. In this paper, we consider matrix approximation by
sampling predefined \emph{blocks} of columns (or rows) from the matrix. We
present an algorithm for sampling useful column blocks and provide novel
guarantees for the quality of the approximation. This algorithm has application
in problems as diverse as biometric data analysis to distributed computing. We
demonstrate the effectiveness of the proposed algorithms for computing the
Block CUR decomposition of large matrices in a distributed setting with
multiple nodes in a compute cluster, where such blocks correspond to columns
(or rows) of the matrix stored on the same node, which can be retrieved with
much less overhead than retrieving individual columns stored across different
nodes. In the biometric setting, the rows correspond to different users and
columns correspond to users' biometric reaction to external stimuli, {\em
e.g.,}~watching video content, at a particular time instant. There is
significant cost in acquiring each user's reaction to lengthy content so we
sample a few important scenes to approximate the biometric response. An
individual time sample in this use case cannot be queried in isolation due to
the lack of context that caused that biometric reaction. Instead, collections
of time segments ({\em i.e.,} blocks) must be presented to the user. The
practical application of these algorithms is shown via experimental results
using real-world user biometric data from a content testing environment.Comment: shorter version to appear in ECML-PKDD 201
A Matrix Hyperbolic Cosine Algorithm and Applications
In this paper, we generalize Spencer's hyperbolic cosine algorithm to the
matrix-valued setting. We apply the proposed algorithm to several problems by
analyzing its computational efficiency under two special cases of matrices; one
in which the matrices have a group structure and an other in which they have
rank-one. As an application of the former case, we present a deterministic
algorithm that, given the multiplication table of a finite group of size ,
it constructs an expanding Cayley graph of logarithmic degree in near-optimal
O(n^2 log^3 n) time. For the latter case, we present a fast deterministic
algorithm for spectral sparsification of positive semi-definite matrices, which
implies an improved deterministic algorithm for spectral graph sparsification
of dense graphs. In addition, we give an elementary connection between spectral
sparsification of positive semi-definite matrices and element-wise matrix
sparsification. As a consequence, we obtain improved element-wise
sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work
in (current) Section
On almost randomizing channels with a short Kraus decomposition
For large d, we study quantum channels on C^d obtained by selecting randomly
N independent Kraus operators according to a probability measure mu on the
unitary group U(d). When mu is the Haar measure, we show that for
N>d/epsilon^2. For d=2^k (k qubits), this includes Kraus operators
obtained by tensoring k random Pauli matrices. The proof uses recent results on
empirical processes in Banach spaces.Comment: We added some background on geometry of Banach space
Structured Random Matrices
Random matrix theory is a well-developed area of probability theory that has
numerous connections with other areas of mathematics and its applications. Much
of the literature in this area is concerned with matrices that possess many
exact or approximate symmetries, such as matrices with i.i.d. entries, for
which precise analytic results and limit theorems are available. Much less well
understood are matrices that are endowed with an arbitrary structure, such as
sparse Wigner matrices or matrices whose entries possess a given variance
pattern. The challenge in investigating such structured random matrices is to
understand how the given structure of the matrix is reflected in its spectral
properties. This chapter reviews a number of recent results, methods, and open
problems in this direction, with a particular emphasis on sharp spectral norm
inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and
Applications" (Springer
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Convex recovery of a structured signal from independent random linear measurements
This chapter develops a theoretical analysis of the convex programming method
for recovering a structured signal from independent random linear measurements.
This technique delivers bounds for the sampling complexity that are similar
with recent results for standard Gaussian measurements, but the argument
applies to a much wider class of measurement ensembles. To demonstrate the
power of this approach, the paper presents a short analysis of phase retrieval
by trace-norm minimization. The key technical tool is a framework, due to
Mendelson and coauthors, for bounding a nonnegative empirical process.Comment: 18 pages, 1 figure. To appear in "Sampling Theory, a Renaissance."
v2: minor corrections. v3: updated citations and increased emphasis on
Mendelson's contribution
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