765 research outputs found
Spectral learning of general weighted automata via constrained matrix completion
Student Paper Awards NIPS 2012Many tasks in text and speech processing and computational biology require estimating
functions mapping strings to real numbers. A broad class of such functions
can be defined by weighted automata. Spectral methods based on the singular
value decomposition of a Hankel matrix have been recently proposed for
learning a probability distribution represented by a weighted automaton from a
training sample drawn according to this same target distribution. In this paper, we
show how spectral methods can be extended to the problem of learning a general
weighted automaton from a sample generated by an arbitrary distribution. The
main obstruction to this approach is that, in general, some entries of the Hankel
matrix may be missing. We present a solution to this problem based on solving a
constrained matrix completion problem. Combining these two ingredients, matrix
completion and spectral method, a whole new family of algorithms for learning
general weighted automata is obtained. We present generalization bounds for a
particular algorithm in this family. The proofs rely on a joint stability analysis of
matrix completion and spectral learning.Peer ReviewedAward-winningPostprint (published version
Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces
We model a problem motivated by road design as a feasibility problem.
Projections onto the constraint sets are obtained, and projection methods for
solving the feasibility problem are studied. We present results of numerical
experiments which demonstrate the efficacy of projection methods even for
challenging nonconvex problems
Signal Decomposition Using Masked Proximal Operators
We consider the well-studied problem of decomposing a vector time series
signal into components with different characteristics, such as smooth,
periodic, nonnegative, or sparse. We describe a simple and general framework in
which the components are defined by loss functions (which include constraints),
and the signal decomposition is carried out by minimizing the sum of losses of
the components (subject to the constraints). When each loss function is the
negative log-likelihood of a density for the signal component, this framework
coincides with maximum a posteriori probability (MAP) estimation; but it also
includes many other interesting cases. Summarizing and clarifying prior
results, we give two distributed optimization methods for computing the
decomposition, which find the optimal decomposition when the component class
loss functions are convex, and are good heuristics when they are not. Both
methods require only the masked proximal operator of each of the component loss
functions, a generalization of the well-known proximal operator that handles
missing entries in its argument. Both methods are distributed, i.e., handle
each component separately. We derive tractable methods for evaluating the
masked proximal operators of some loss functions that, to our knowledge, have
not appeared in the literature.Comment: The manuscript has 61 pages, 22 figures and 2 tables. Also hosted at
https://web.stanford.edu/~boyd/papers/sig_decomp_mprox.html. For code, see
https://github.com/cvxgrp/signal-decompositio
Towards a theory of patches
AbstractMany applications have a need for indexing unstructured data. It turns out that a similar ad-hoc method is being used in many of them – that of considering small particles of the data.In this paper we formalize this concept as a tiling problem and consider the efficiency of dealing with this model in the pattern matching setting.We present an efficient algorithm for the one-dimensional tiling problem, and the one-dimensional tiled pattern matching problem. We prove the two-dimensional problem is hard and then develop an approximation algorithm with an approximation ratio converging to 2. We show that other two-dimensional versions of the problem are also hard, regardless of the number of neighbors a tile has
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