129,156 research outputs found
Low Rank Directed Acyclic Graphs and Causal Structure Learning
Despite several important advances in recent years, learning causal
structures represented by directed acyclic graphs (DAGs) remains a challenging
task in high dimensional settings when the graphs to be learned are not sparse.
In particular, the recent formulation of structure learning as a continuous
optimization problem proved to have considerable advantages over the
traditional combinatorial formulation, but the performance of the resulting
algorithms is still wanting when the target graph is relatively large and
dense. In this paper we propose a novel approach to mitigate this problem, by
exploiting a low rank assumption regarding the (weighted) adjacency matrix of a
DAG causal model. We establish several useful results relating interpretable
graphical conditions to the low rank assumption, and show how to adapt existing
methods for causal structure learning to take advantage of this assumption. We
also provide empirical evidence for the utility of our low rank algorithms,
especially on graphs that are not sparse. Not only do they outperform
state-of-the-art algorithms when the low rank condition is satisfied, the
performance on randomly generated scale-free graphs is also very competitive
even though the true ranks may not be as low as is assumed
Matrix Completion on Graphs
The problem of finding the missing values of a matrix given a few of its
entries, called matrix completion, has gathered a lot of attention in the
recent years. Although the problem under the standard low rank assumption is
NP-hard, Cand\`es and Recht showed that it can be exactly relaxed if the number
of observed entries is sufficiently large. In this work, we introduce a novel
matrix completion model that makes use of proximity information about rows and
columns by assuming they form communities. This assumption makes sense in
several real-world problems like in recommender systems, where there are
communities of people sharing preferences, while products form clusters that
receive similar ratings. Our main goal is thus to find a low-rank solution that
is structured by the proximities of rows and columns encoded by graphs. We
borrow ideas from manifold learning to constrain our solution to be smooth on
these graphs, in order to implicitly force row and column proximities. Our
matrix recovery model is formulated as a convex non-smooth optimization
problem, for which a well-posed iterative scheme is provided. We study and
evaluate the proposed matrix completion on synthetic and real data, showing
that the proposed structured low-rank recovery model outperforms the standard
matrix completion model in many situations.Comment: Version of NIPS 2014 workshop "Out of the Box: Robustness in High
Dimension
Subspace Expanders and Matrix Rank Minimization
Matrix rank minimization (RM) problems recently gained extensive attention
due to numerous applications in machine learning, system identification and
graphical models. In RM problem, one aims to find the matrix with the lowest
rank that satisfies a set of linear constraints. The existing algorithms
include nuclear norm minimization (NNM) and singular value thresholding. Thus
far, most of the attention has been on i.i.d. Gaussian measurement operators.
In this work, we introduce a new class of measurement operators, and a novel
recovery algorithm, which is notably faster than NNM. The proposed operators
are based on what we refer to as subspace expanders, which are inspired by the
well known expander graphs based measurement matrices in compressed sensing. We
show that given an PSD matrix of rank , it can be uniquely
recovered from a minimal sampling of measurements using the proposed
structures, and the recovery algorithm can be cast as matrix inversion after a
few initial processing steps
Unsupervised Graph-based Rank Aggregation for Improved Retrieval
This paper presents a robust and comprehensive graph-based rank aggregation
approach, used to combine results of isolated ranker models in retrieval tasks.
The method follows an unsupervised scheme, which is independent of how the
isolated ranks are formulated. Our approach is able to combine arbitrary
models, defined in terms of different ranking criteria, such as those based on
textual, image or hybrid content representations.
We reformulate the ad-hoc retrieval problem as a document retrieval based on
fusion graphs, which we propose as a new unified representation model capable
of merging multiple ranks and expressing inter-relationships of retrieval
results automatically. By doing so, we claim that the retrieval system can
benefit from learning the manifold structure of datasets, thus leading to more
effective results. Another contribution is that our graph-based aggregation
formulation, unlike existing approaches, allows for encapsulating contextual
information encoded from multiple ranks, which can be directly used for
ranking, without further computations and post-processing steps over the
graphs. Based on the graphs, a novel similarity retrieval score is formulated
using an efficient computation of minimum common subgraphs. Finally, another
benefit over existing approaches is the absence of hyperparameters.
A comprehensive experimental evaluation was conducted considering diverse
well-known public datasets, composed of textual, image, and multimodal
documents. Performed experiments demonstrate that our method reaches top
performance, yielding better effectiveness scores than state-of-the-art
baseline methods and promoting large gains over the rankers being fused, thus
demonstrating the successful capability of the proposal in representing queries
based on a unified graph-based model of rank fusions
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