27,779 research outputs found
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
Statistically Motivated Second Order Pooling
Second-order pooling, a.k.a.~bilinear pooling, has proven effective for deep
learning based visual recognition. However, the resulting second-order networks
yield a final representation that is orders of magnitude larger than that of
standard, first-order ones, making them memory-intensive and cumbersome to
deploy. Here, we introduce a general, parametric compression strategy that can
produce more compact representations than existing compression techniques, yet
outperform both compressed and uncompressed second-order models. Our approach
is motivated by a statistical analysis of the network's activations, relying on
operations that lead to a Gaussian-distributed final representation, as
inherently used by first-order deep networks. As evidenced by our experiments,
this lets us outperform the state-of-the-art first-order and second-order
models on several benchmark recognition datasets.Comment: Accepted to ECCV 2018. Camera ready version. 14 page, 5 figures, 3
table
A generalized topological recursion for arbitrary ramification
The Eynard-Orantin topological recursion relies on the geometry of a Riemann
surface S and two meromorphic functions x and y on S. To formulate the
recursion, one must assume that x has only simple ramification points. In this
paper we propose a generalized topological recursion that is valid for x with
arbitrary ramification. We justify our proposal by studying degenerations of
Riemann surfaces. We check in various examples that our generalized recursion
is compatible with invariance of the free energies under the transformation
(x,y) -> (y,x), where either x or y (or both) have higher order ramification,
and that it satisfies some of the most important properties of the original
recursion. Along the way, we show that invariance under (x,y) -> (y,x) is in
fact more subtle than expected; we show that there exists a number of counter
examples, already in the case of the original Eynard-Orantin recursion, that
deserve further study.Comment: 26 pages, 2 figure
Topological Optimization of the Evaluation of Finite Element Matrices
We present a topological framework for finding low-flop algorithms for
evaluating element stiffness matrices associated with multilinear forms for
finite element methods posed over straight-sided affine domains. This framework
relies on phrasing the computation on each element as the contraction of each
collection of reference element tensors with an element-specific geometric
tensor. We then present a new concept of complexity-reducing relations that
serve as distance relations between these reference element tensors. This
notion sets up a graph-theoretic context in which we may find an optimized
algorithm by computing a minimum spanning tree. We present experimental results
for some common multilinear forms showing significant reductions in operation
count and also discuss some efficient algorithms for building the graph we use
for the optimization
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