7,350 research outputs found
Lower Bounds for Matrix Factorization
We study the problem of constructing explicit families of matrices which
cannot be expressed as a product of a few sparse matrices. In addition to being
a natural mathematical question on its own, this problem appears in various
incarnations in computer science; the most significant being in the context of
lower bounds for algebraic circuits which compute linear transformations,
matrix rigidity and data structure lower bounds.
We first show, for every constant , a deterministic construction in
subexponential time of a family of matrices which cannot
be expressed as a product where the total sparsity of
is less than . In other words, any depth-
linear circuit computing the linear transformation has size at
least . This improves upon the prior best lower bounds for
this problem, which are barely super-linear, and were obtained by a long line
of research based on the study of super-concentrators (albeit at the cost of a
blow up in the time required to construct these matrices).
We then outline an approach for proving improved lower bounds through a
certain derandomization problem, and use this approach to prove asymptotically
optimal quadratic lower bounds for natural special cases, which generalize many
of the common matrix decompositions
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
Improved rank bounds for design matrices and a new proof of Kelly's theorem
We study the rank of complex sparse matrices in which the supports of
different columns have small intersections. The rank of these matrices, called
design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in
which they were used to answer questions regarding point configurations. In
this work we derive near-optimal rank bounds for these matrices and use them to
obtain asymptotically tight bounds in many of the geometric applications. As a
consequence of our improved analysis, we also obtain a new, linear algebraic,
proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai
theorem
On analog quantum algorithms for the mixing of Markov chains
The problem of sampling from the stationary distribution of a Markov chain
finds widespread applications in a variety of fields. The time required for a
Markov chain to converge to its stationary distribution is known as the
classical mixing time. In this article, we deal with analog quantum algorithms
for mixing. First, we provide an analog quantum algorithm that given a Markov
chain, allows us to sample from its stationary distribution in a time that
scales as the sum of the square root of the classical mixing time and the
square root of the classical hitting time. Our algorithm makes use of the
framework of interpolated quantum walks and relies on Hamiltonian evolution in
conjunction with von Neumann measurements.
There also exists a different notion for quantum mixing: the problem of
sampling from the limiting distribution of quantum walks, defined in a
time-averaged sense. In this scenario, the quantum mixing time is defined as
the time required to sample from a distribution that is close to this limiting
distribution. Recently we provided an upper bound on the quantum mixing time
for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we
also extend and expand upon our findings therein. Namely, we provide an
intuitive understanding of the state-of-the-art random matrix theory tools used
to derive our results. In particular, for our analysis we require information
about macroscopic, mesoscopic and microscopic statistics of eigenvalues of
random matrices which we highlight here. Furthermore, we provide numerical
simulations that corroborate our analytical findings and extend this notion of
mixing from simple graphs to any ergodic, reversible, Markov chain.Comment: The section concerning time-averaged mixing (Sec VIII) has been
updated: Now contains numerical plots and an intuitive discussion on the
random matrix theory results used to derive the results of arXiv:2001.0630
Universality for general Wigner-type matrices
We consider the local eigenvalue distribution of large self-adjoint random matrices with centered independent entries.
In contrast to previous works the matrix of variances is not assumed to be stochastic. Hence the density of states is
not the Wigner semicircle law. Its possible shapes are described in the
companion paper [1]. We show that as grows, the resolvent,
, converges to a diagonal matrix, , where
solves the vector equation that has
been analyzed in [1]. We prove a local law down to the smallest spectral
resolution scale, and bulk universality for both real symmetric and complex
hermitian symmetry classes.Comment: Changes in version 3: The format of pictures was changed to resolve a
conflict with certain pdf viewer
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