743 research outputs found
Tensor Graphical Lasso (TeraLasso)
This paper introduces a multi-way tensor generalization of the Bigraphical
Lasso (BiGLasso), which uses a two-way sparse Kronecker-sum multivariate-normal
model for the precision matrix to parsimoniously model conditional dependence
relationships of matrix-variate data based on the Cartesian product of graphs.
We call this generalization the {\bf Te}nsor g{\bf ra}phical Lasso (TeraLasso).
We demonstrate using theory and examples that the TeraLasso model can be
accurately and scalably estimated from very limited data samples of high
dimensional variables with multiway coordinates such as space, time and
replicates. Statistical consistency and statistical rates of convergence are
established for both the BiGLasso and TeraLasso estimators of the precision
matrix and estimators of its support (non-sparsity) set, respectively. We
propose a scalable composite gradient descent algorithm and analyze the
computational convergence rate, showing that the composite gradient descent
algorithm is guaranteed to converge at a geometric rate to the global minimizer
of the TeraLasso objective function. Finally, we illustrate the TeraLasso using
both simulation and experimental data from a meteorological dataset, showing
that we can accurately estimate precision matrices and recover meaningful
conditional dependency graphs from high dimensional complex datasets.Comment: accepted to JRSS-
Semi-supervised tensor-based graph embedding learning and its application to visual discriminant tracking
An appearance model adaptable to changes in object appearance is critical in visual object tracking. In
this paper, we treat an image patch as a 2-order tensor which preserves the original image structure. We design
two graphs for characterizing the intrinsic local geometrical structure of the tensor samples of the object and the
background. Graph embedding is used to reduce the dimensions of the tensors while preserving the structure of
the graphs. Then, a discriminant embedding space is constructed. We prove two propositions for finding the
transformation matrices which are used to map the original tensor samples to the tensor-based graph embedding
space. In order to encode more discriminant information in the embedding space, we propose a transfer-learningbased
semi-supervised strategy to iteratively adjust the embedding space into which discriminative information
obtained from earlier times is transferred. We apply the proposed semi-supervised tensor-based graph
embedding learning algorithm to visual tracking. The new tracking algorithm captures an object’s appearance
characteristics during tracking and uses a particle filter to estimate the optimal object state. Experimental results
on the CVPR 2013 benchmark dataset demonstrate the effectiveness of the proposed tracking algorithm
Tight bounds on the convergence rate of generalized ratio consensus algorithms
The problems discussed in this paper are motivated by general ratio consensus
algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as
the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the
name weighted gossip algorithm. We consider a communication protocol described
by a strictly stationary, ergodic, sequentially primitive sequence of
non-negative matrices, applied iteratively to a pair of fixed initial vectors,
the components of which are called values and weights defined at the nodes of a
network. The subject of ratio consensus problems is to study the asymptotic
properties of ratios of values and weights at each node, expecting convergence
to the same limit for all nodes. The main results of the paper provide upper
bounds for the rate of the almost sure exponential convergence in terms of the
spectral gap associated with the given sequence of random matrices. It will be
shown that these upper bounds are sharp. Our results complement previous
results of Picci and Taylor (2013) and Iutzeler, Ciblat and Hachem (2013)
Classification of hyperspectral images by tensor modeling and additive morphological decomposition
International audiencePixel-wise classification in high-dimensional multivariate images is investigated. The proposed method deals with the joint use of spectral and spatial information provided in hyperspectral images. Additive morphological decomposition (AMD) based on morphological operators is proposed. AMD defines a scale-space decomposition for multivariate images without any loss of information. AMD is modeled as a tensor structure and tensor principal components analysis is compared as dimensional reduction algorithm versus classic approach. Experimental comparison shows that the proposed algorithm can provide better performance for the pixel classification of hyperspectral image than many other well-known techniques
Geometric representations for minimalist grammars
We reformulate minimalist grammars as partial functions on term algebras for
strings and trees. Using filler/role bindings and tensor product
representations, we construct homomorphisms for these data structures into
geometric vector spaces. We prove that the structure-building functions as well
as simple processors for minimalist languages can be realized by piecewise
linear operators in representation space. We also propose harmony, i.e. the
distance of an intermediate processing step from the final well-formed state in
representation space, as a measure of processing complexity. Finally, we
illustrate our findings by means of two particular arithmetic and fractal
representations.Comment: 43 pages, 4 figure
Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off
Kernel methods are powerful learning methodologies that provide a simple way
to construct nonlinear algorithms from linear ones. Despite their popularity,
they suffer from poor scalability in big data scenarios. Various approximation
methods, including random feature approximation have been proposed to alleviate
the problem. However, the statistical consistency of most of these approximate
kernel methods is not well understood except for kernel ridge regression
wherein it has been shown that the random feature approximation is not only
computationally efficient but also statistically consistent with a minimax
optimal rate of convergence. In this paper, we investigate the efficacy of
random feature approximation in the context of kernel principal component
analysis (KPCA) by studying the trade-off between computational and statistical
behaviors of approximate KPCA. We show that the approximate KPCA is both
computationally and statistically efficient compared to KPCA in terms of the
error associated with reconstructing a kernel function based on its projection
onto the corresponding eigenspaces. Depending on the eigenvalue decay behavior
of the covariance operator, we show that only features (polynomial
decay) or features (exponential decay) are needed to match the
statistical performance of KPCA. We also investigate their statistical
behaviors in terms of the convergence of corresponding eigenspaces wherein we
show that only features are required to match the performance of
KPCA and if fewer than features are used, then approximate KPCA has
a worse statistical behavior than that of KPCA.Comment: 46 page
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