415 research outputs found
Spatially Aware Dictionary Learning and Coding for Fossil Pollen Identification
We propose a robust approach for performing automatic species-level
recognition of fossil pollen grains in microscopy images that exploits both
global shape and local texture characteristics in a patch-based matching
methodology. We introduce a novel criteria for selecting meaningful and
discriminative exemplar patches. We optimize this function during training
using a greedy submodular function optimization framework that gives a
near-optimal solution with bounded approximation error. We use these selected
exemplars as a dictionary basis and propose a spatially-aware sparse coding
method to match testing images for identification while maintaining global
shape correspondence. To accelerate the coding process for fast matching, we
introduce a relaxed form that uses spatially-aware soft-thresholding during
coding. Finally, we carry out an experimental study that demonstrates the
effectiveness and efficiency of our exemplar selection and classification
mechanisms, achieving accuracy on a difficult fine-grained species
classification task distinguishing three types of fossil spruce pollen.Comment: CVMI 201
Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection
We study the problem of selecting a subset of k random variables from a large
set, in order to obtain the best linear prediction of another variable of
interest. This problem can be viewed in the context of both feature selection
and sparse approximation. We analyze the performance of widely used greedy
heuristics, using insights from the maximization of submodular functions and
spectral analysis. We introduce the submodularity ratio as a key quantity to
help understand why greedy algorithms perform well even when the variables are
highly correlated. Using our techniques, we obtain the strongest known
approximation guarantees for this problem, both in terms of the submodularity
ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further
demonstrate the wide applicability of our techniques by analyzing greedy
algorithms for the dictionary selection problem, and significantly improve the
previously known guarantees. Our theoretical analysis is complemented by
experiments on real-world and synthetic data sets; the experiments show that
the submodularity ratio is a stronger predictor of the performance of greedy
algorithms than other spectral parameters
Structured sparsity-inducing norms through submodular functions
Sparse methods for supervised learning aim at finding good linear predictors
from as few variables as possible, i.e., with small cardinality of their
supports. This combinatorial selection problem is often turned into a convex
optimization problem by replacing the cardinality function by its convex
envelope (tightest convex lower bound), in this case the L1-norm. In this
paper, we investigate more general set-functions than the cardinality, that may
incorporate prior knowledge or structural constraints which are common in many
applications: namely, we show that for nondecreasing submodular set-functions,
the corresponding convex envelope can be obtained from its \lova extension, a
common tool in submodular analysis. This defines a family of polyhedral norms,
for which we provide generic algorithmic tools (subgradients and proximal
operators) and theoretical results (conditions for support recovery or
high-dimensional inference). By selecting specific submodular functions, we can
give a new interpretation to known norms, such as those based on
rank-statistics or grouped norms with potentially overlapping groups; we also
define new norms, in particular ones that can be used as non-factorial priors
for supervised learning
Structured Sparsity: Discrete and Convex approaches
Compressive sensing (CS) exploits sparsity to recover sparse or compressible
signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity
is also used to enhance interpretability in machine learning and statistics
applications: While the ambient dimension is vast in modern data analysis
problems, the relevant information therein typically resides in a much lower
dimensional space. However, many solutions proposed nowadays do not leverage
the true underlying structure. Recent results in CS extend the simple sparsity
idea to more sophisticated {\em structured} sparsity models, which describe the
interdependency between the nonzero components of a signal, allowing to
increase the interpretability of the results and lead to better recovery
performance. In order to better understand the impact of structured sparsity,
in this chapter we analyze the connections between the discrete models and
their convex relaxations, highlighting their relative advantages. We start with
the general group sparse model and then elaborate on two important special
cases: the dispersive and the hierarchical models. For each, we present the
models in their discrete nature, discuss how to solve the ensuing discrete
problems and then describe convex relaxations. We also consider more general
structures as defined by set functions and present their convex proxies.
Further, we discuss efficient optimization solutions for structured sparsity
problems and illustrate structured sparsity in action via three applications.Comment: 30 pages, 18 figure
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