1,291 research outputs found
Inferring Rankings Using Constrained Sensing
We consider the problem of recovering a function over the space of
permutations (or, the symmetric group) over elements from given partial
information; the partial information we consider is related to the group
theoretic Fourier Transform of the function. This problem naturally arises in
several settings such as ranked elections, multi-object tracking, ranking
systems, and recommendation systems. Inspired by the work of Donoho and Stark
in the context of discrete-time functions, we focus on non-negative functions
with a sparse support (support size domain size). Our recovery method is
based on finding the sparsest solution (through optimization) that is
consistent with the available information. As the main result, we derive
sufficient conditions for functions that can be recovered exactly from partial
information through optimization. Under a natural random model for the
generation of functions, we quantify the recoverability conditions by deriving
bounds on the sparsity (support size) for which the function satisfies the
sufficient conditions with a high probability as .
optimization is computationally hard. Therefore, the popular compressive
sensing literature considers solving the convex relaxation,
optimization, to find the sparsest solution. However, we show that
optimization fails to recover a function (even with constant sparsity)
generated using the random model with a high probability as . In
order to overcome this problem, we propose a novel iterative algorithm for the
recovery of functions that satisfy the sufficient conditions. Finally, using an
Information Theoretic framework, we study necessary conditions for exact
recovery to be possible.Comment: 19 page
Compressive Network Analysis
Modern data acquisition routinely produces massive amounts of network data.
Though many methods and models have been proposed to analyze such data, the
research of network data is largely disconnected with the classical theory of
statistical learning and signal processing. In this paper, we present a new
framework for modeling network data, which connects two seemingly different
areas: network data analysis and compressed sensing. From a nonparametric
perspective, we model an observed network using a large dictionary. In
particular, we consider the network clique detection problem and show
connections between our formulation with a new algebraic tool, namely Randon
basis pursuit in homogeneous spaces. Such a connection allows us to identify
rigorous recovery conditions for clique detection problems. Though this paper
is mainly conceptual, we also develop practical approximation algorithms for
solving empirical problems and demonstrate their usefulness on real-world
datasets
Minimax-optimal Inference from Partial Rankings
This paper studies the problem of inferring a global preference based on the
partial rankings provided by many users over different subsets of items
according to the Plackett-Luce model. A question of particular interest is how
to optimally assign items to users for ranking and how many item assignments
are needed to achieve a target estimation error. For a given assignment of
items to users, we first derive an oracle lower bound of the estimation error
that holds even for the more general Thurstone models. Then we show that the
Cram\'er-Rao lower bound and our upper bounds inversely depend on the spectral
gap of the Laplacian of an appropriately defined comparison graph. When the
system is allowed to choose the item assignment, we propose a random assignment
scheme. Our oracle lower bound and upper bounds imply that it is
minimax-optimal up to a logarithmic factor among all assignment schemes and the
lower bound can be achieved by the maximum likelihood estimator as well as
popular rank-breaking schemes that decompose partial rankings into pairwise
comparisons. The numerical experiments corroborate our theoretical findings.Comment: 16 pages, 2 figure
Summary Based Structures with Improved Sublinear Recovery for Compressed Sensing
We introduce a new class of measurement matrices for compressed sensing,
using low order summaries over binary sequences of a given length. We prove
recovery guarantees for three reconstruction algorithms using the proposed
measurements, including minimization and two combinatorial methods. In
particular, one of the algorithms recovers -sparse vectors of length in
sublinear time , and requires at most
measurements. The empirical oversampling constant
of the algorithm is significantly better than existing sublinear recovery
algorithms such as Chaining Pursuit and Sudocodes. In particular, for and , the oversampling factor is between 3 to 8. We provide
preliminary insight into how the proposed constructions, and the fast recovery
scheme can be used in a number of practical applications such as market basket
analysis, and real time compressed sensing implementation
On Estimating Multi-Attribute Choice Preferences using Private Signals and Matrix Factorization
Revealed preference theory studies the possibility of modeling an agent's
revealed preferences and the construction of a consistent utility function.
However, modeling agent's choices over preference orderings is not always
practical and demands strong assumptions on human rationality and
data-acquisition abilities. Therefore, we propose a simple generative choice
model where agents are assumed to generate the choice probabilities based on
latent factor matrices that capture their choice evaluation across multiple
attributes. Since the multi-attribute evaluation is typically hidden within the
agent's psyche, we consider a signaling mechanism where agents are provided
with choice information through private signals, so that the agent's choices
provide more insight about his/her latent evaluation across multiple
attributes. We estimate the choice model via a novel multi-stage matrix
factorization algorithm that minimizes the average deviation of the factor
estimates from choice data. Simulation results are presented to validate the
estimation performance of our proposed algorithm.Comment: 6 pages, 2 figures, to be presented at CISS conferenc
A Topic Modeling Approach to Ranking
We propose a topic modeling approach to the prediction of preferences in
pairwise comparisons. We develop a new generative model for pairwise
comparisons that accounts for multiple shared latent rankings that are
prevalent in a population of users. This new model also captures inconsistent
user behavior in a natural way. We show how the estimation of latent rankings
in the new generative model can be formally reduced to the estimation of topics
in a statistically equivalent topic modeling problem. We leverage recent
advances in the topic modeling literature to develop an algorithm that can
learn shared latent rankings with provable consistency as well as sample and
computational complexity guarantees. We demonstrate that the new approach is
empirically competitive with the current state-of-the-art approaches in
predicting preferences on some semi-synthetic and real world datasets
Inferring rankings using constrained sensing
We consider the problem of recovering a function over the space of permutations (or, the symmetric group) over n elements from given partial information; the partial information we consider is related to the group theoretic Fourier Transform of the function. This problem naturally arises in several settings such as ranked elections, multi-object tracking, ranking systems, and recommendation systems. Inspired by the work of Donoho and Stark in the context of discrete-time functions, we focus on non-negative functions with a sparse support (support size <;<; domain size). Our recovery method is based on finding the sparsest solution (through l[subscript 0] optimization) that is consistent with the available information. As the main result, we derive sufficient conditions for functions that can be recovered exactly from partial information through l[subscript 0] optimization. Under a natural random model for the generation of functions, we quantify the recoverability conditions by deriving bounds on the sparsity (support size) for which the function satisfies the sufficient conditions with a high probability as n β β. β0 optimization is computationally hard. Therefore, the popular compressive sensing literature considers solving the convex relaxation, β[subscript 1] optimization, to find the sparsest solution. However, we show that β[subscript 1] optimization fails to recover a function (even with constant sparsity) generated using the random model with a high probability as n β β. In order to overcome this problem, we propose a novel iterative algorithm for the recovery of functions that satisfy the sufficient conditions. Finally, using an Information Theoretic framework, we study necessary conditions for exact recovery to be possible
KCRC-LCD: Discriminative Kernel Collaborative Representation with Locality Constrained Dictionary for Visual Categorization
We consider the image classification problem via kernel collaborative
representation classification with locality constrained dictionary (KCRC-LCD).
Specifically, we propose a kernel collaborative representation classification
(KCRC) approach in which kernel method is used to improve the discrimination
ability of collaborative representation classification (CRC). We then measure
the similarities between the query and atoms in the global dictionary in order
to construct a locality constrained dictionary (LCD) for KCRC. In addition, we
discuss several similarity measure approaches in LCD and further present a
simple yet effective unified similarity measure whose superiority is validated
in experiments. There are several appealing aspects associated with LCD. First,
LCD can be nicely incorporated under the framework of KCRC. The LCD similarity
measure can be kernelized under KCRC, which theoretically links CRC and LCD
under the kernel method. Second, KCRC-LCD becomes more scalable to both the
training set size and the feature dimension. Example shows that KCRC is able to
perfectly classify data with certain distribution, while conventional CRC fails
completely. Comprehensive experiments on many public datasets also show that
KCRC-LCD is a robust discriminative classifier with both excellent performance
and good scalability, being comparable or outperforming many other
state-of-the-art approaches
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