8,832 research outputs found
Greedy Shallow Networks: An Approach for Constructing and Training Neural Networks
We present a greedy-based approach to construct an efficient single hidden
layer neural network with the ReLU activation that approximates a target
function. In our approach we obtain a shallow network by utilizing a greedy
algorithm with the prescribed dictionary provided by the available training
data and a set of possible inner weights. To facilitate the greedy selection
process we employ an integral representation of the network, based on the
ridgelet transform, that significantly reduces the cardinality of the
dictionary and hence promotes feasibility of the greedy selection. Our approach
allows for the construction of efficient architectures which can be treated
either as improved initializations to be used in place of random-based
alternatives, or as fully-trained networks in certain cases, thus potentially
nullifying the need for backpropagation training. Numerical experiments
demonstrate the tenability of the proposed concept and its advantages compared
to the conventional techniques for selecting architectures and initializations
for neural networks
Sparse Image Representation with Epitomes
Sparse coding, which is the decomposition of a vector using only a few basis
elements, is widely used in machine learning and image processing. The basis
set, also called dictionary, is learned to adapt to specific data. This
approach has proven to be very effective in many image processing tasks.
Traditionally, the dictionary is an unstructured "flat" set of atoms. In this
paper, we study structured dictionaries which are obtained from an epitome, or
a set of epitomes. The epitome is itself a small image, and the atoms are all
the patches of a chosen size inside this image. This considerably reduces the
number of parameters to learn and provides sparse image decompositions with
shiftinvariance properties. We propose a new formulation and an algorithm for
learning the structured dictionaries associated with epitomes, and illustrate
their use in image denoising tasks.Comment: Computer Vision and Pattern Recognition, Colorado Springs : United
States (2011
Learning Sparsely Used Overcomplete Dictionaries via Alternating Minimization
We consider the problem of sparse coding, where each sample consists of a
sparse linear combination of a set of dictionary atoms, and the task is to
learn both the dictionary elements and the mixing coefficients. Alternating
minimization is a popular heuristic for sparse coding, where the dictionary and
the coefficients are estimated in alternate steps, keeping the other fixed.
Typically, the coefficients are estimated via minimization, keeping
the dictionary fixed, and the dictionary is estimated through least squares,
keeping the coefficients fixed. In this paper, we establish local linear
convergence for this variant of alternating minimization and establish that the
basin of attraction for the global optimum (corresponding to the true
dictionary and the coefficients) is \order{1/s^2}, where is the sparsity
level in each sample and the dictionary satisfies RIP. Combined with the recent
results of approximate dictionary estimation, this yields provable guarantees
for exact recovery of both the dictionary elements and the coefficients, when
the dictionary elements are incoherent.Comment: Local linear convergence now holds under RIP and also more general
restricted eigenvalue condition
Flexible Multi-layer Sparse Approximations of Matrices and Applications
The computational cost of many signal processing and machine learning
techniques is often dominated by the cost of applying certain linear operators
to high-dimensional vectors. This paper introduces an algorithm aimed at
reducing the complexity of applying linear operators in high dimension by
approximately factorizing the corresponding matrix into few sparse factors. The
approach relies on recent advances in non-convex optimization. It is first
explained and analyzed in details and then demonstrated experimentally on
various problems including dictionary learning for image denoising, and the
approximation of large matrices arising in inverse problems
- …