193 research outputs found
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
An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure
We present an exploration of the rich theoretical connections between several
classes of regularized models, network flows, and recent results in submodular
function theory. This work unifies key aspects of these problems under a common
theory, leading to novel methods for working with several important models of
interest in statistics, machine learning and computer vision.
In Part 1, we review the concepts of network flows and submodular function
optimization theory foundational to our results. We then examine the
connections between network flows and the minimum-norm algorithm from
submodular optimization, extending and improving several current results. This
leads to a concise representation of the structure of a large class of pairwise
regularized models important in machine learning, statistics and computer
vision.
In Part 2, we describe the full regularization path of a class of penalized
regression problems with dependent variables that includes the graph-guided
LASSO and total variation constrained models. This description also motivates a
practical algorithm. This allows us to efficiently find the regularization path
of the discretized version of TV penalized models. Ultimately, our new
algorithms scale up to high-dimensional problems with millions of variables
Greedy Bayesian Posterior Approximation with Deep Ensembles
Ensembles of independently trained neural networks are a state-of-the-art
approach to estimate predictive uncertainty in Deep Learning, and can be
interpreted as an approximation of the posterior distribution via a mixture of
delta functions. The training of ensembles relies on non-convexity of the loss
landscape and random initialization of their individual members, making the
resulting posterior approximation uncontrolled. This paper proposes a novel and
principled method to tackle this limitation, minimizing an -divergence
between the true posterior and a kernel density estimator in a function space.
We analyze this objective from a combinatorial point of view, and show that it
is submodular with respect to mixture components for any . Subsequently, we
consider the problem of ensemble construction, and from the marginal gain of
the total objective, we derive a novel diversity term for training ensembles
greedily. The performance of our approach is demonstrated on computer vision
out-of-distribution detection benchmarks in a range of architectures trained on
multiple datasets. The source code of our method is publicly available at
https://github.com/MIPT-Oulu/greedy_ensembles_training
Efficient Sensor Placement from Regression with Sparse Gaussian Processes in Continuous and Discrete Spaces
We present a novel approach based on sparse Gaussian processes (SGPs) to
address the sensor placement problem for monitoring spatially (or
spatiotemporally) correlated phenomena such as temperature and precipitation.
Existing Gaussian process (GP) based sensor placement approaches use GPs with
known kernel function parameters to model a phenomenon and subsequently
optimize the sensor locations in a discretized representation of the
environment. In our approach, we fit an SGP with known kernel function
parameters to randomly sampled unlabeled locations in the environment and show
that the learned inducing points of the SGP inherently solve the sensor
placement problem in continuous spaces. Using SGPs avoids discretizing the
environment and reduces the computation cost from cubic to linear complexity.
When restricted to a candidate set of sensor placement locations, we can use
greedy sequential selection algorithms on the SGP's optimization bound to find
good solutions. We also present an approach to efficiently map our continuous
space solutions to discrete solution spaces using the assignment problem, which
gives us discrete sensor placements optimized in unison. Moreover, we
generalize our approach to model sensors with non-point field-of-view and
integrated observations by leveraging the inherent properties of GPs and SGPs.
Our experimental results on three real-world datasets show that our approaches
generate solution placements that result in reconstruction quality that is
consistently on par or better than the prior state-of-the-art approach while
being significantly faster. Our computationally efficient approaches will
enable both large-scale sensor placement, and fast sensor placement for
informative path planning problems.Comment: 11 pages, 4 figures, preprint, supplementar
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