39,959 research outputs found
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
Exploratory Analysis of Functional Data via Clustering and Optimal Segmentation
We propose in this paper an exploratory analysis algorithm for functional
data. The method partitions a set of functions into clusters and represents
each cluster by a simple prototype (e.g., piecewise constant). The total number
of segments in the prototypes, , is chosen by the user and optimally
distributed among the clusters via two dynamic programming algorithms. The
practical relevance of the method is shown on two real world datasets
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
On multiple simple recourse models
We consider multiple simple recourse (MSR) models, both continuous and integer versions, which generalize the corresponding simple recourse (SR) models by allowing for a refined penalty cost structure for individual shortages and surpluses. It will be shown that (convex approximations of) such MSR models can be represented as explicitly specified continuous SR models, and thus can be solved efficiently by existing algorithms.
Sublabel-Accurate Relaxation of Nonconvex Energies
We propose a novel spatially continuous framework for convex relaxations
based on functional lifting. Our method can be interpreted as a
sublabel-accurate solution to multilabel problems. We show that previously
proposed functional lifting methods optimize an energy which is linear between
two labels and hence require (often infinitely) many labels for a faithful
approximation. In contrast, the proposed formulation is based on a piecewise
convex approximation and therefore needs far fewer labels. In comparison to
recent MRF-based approaches, our method is formulated in a spatially continuous
setting and shows less grid bias. Moreover, in a local sense, our formulation
is the tightest possible convex relaxation. It is easy to implement and allows
an efficient primal-dual optimization on GPUs. We show the effectiveness of our
approach on several computer vision problems
Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review
The paper characterizes classes of functions for which deep learning can be
exponentially better than shallow learning. Deep convolutional networks are a
special case of these conditions, though weight sharing is not the main reason
for their exponential advantage
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