Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInferring Sparsity: Compressed Sensing using Generalized Restricted Boltzmann Machines
108
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this work, we consider compressed sensing reconstruction from $M$ measurements of $K$-sparse structured signals which do not possess a writable correlation model. Assuming that a generative statistical model, such as a Boltzmann machine, can be trained in an unsupervised manner on example signals, we demonstrate how this signal model can be used within a Bayesian framework of signal reconstruction. By deriving a message-passing inference for general distribution restricted Boltzmann machines, we are able to integrate these inferred signal models into approximate message passing for compressed sensing reconstruction. Finally, we show for the MNIST dataset that this approach can be very effective, even for $M < K$.
rate research
Read More
Extracting automatically the complex set of features composing real high-dimensional data is crucial for achieving high performance in machine--learning tasks. Restricted Boltzmann Machines (RBM) are empirically known to be efficient for this purpose, and to be able to generate distributed and graded representations of the data. We characterize the structural conditions (sparsity of the weights, low effective temperature, nonlinearities in the activation functions of hidden units, and adaptation of fields maintaining the activity in the visible layer) allowing RBM to operate in such a compositional phase. Evidence is provided by the replica analysis of an adequate statistical ensemble of random RBMs and by RBM trained on the handwritten digits dataset MNIST.
Calculating the spectral function of two dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two dimensional systems present insurmountable hurdles, ranging from the sign problem in quantum Monte Carlo (MC), to the entanglement area law in tensor network based methods. We hereby present a variational approach based on a Chebyshev expansion of the spectral function and a neural network representation for the wave functions. The Chebyshev moments are obtained by recursively applying the Hamiltonian and projecting on the space of variational states using a modified natural gradient descent method. We compare this approach with a modified approximation of the spectral function which uses a Krylov subspace constructed from the Chebyshev wave-functions. We present results for the one-dimensional and two-dimensional Heisenberg model on the square lattice, and compare to those obtained by other methods in the literature.
Restricted Boltzmann machines (RBMs) are energy-based neural-networks which are commonly used as the building blocks for deep architectures neural architectures. In this work, we derive a deterministic framework for the training, evaluation, and use of RBMs based upon the Thouless-Anderson-Palmer (TAP) mean-field approximation of widely-connected systems with weak interactions coming from spin-glass theory. While the TAP approach has been extensively studied for fully-visible binary spin systems, our construction is generalized to latent-variable models, as well as to arbitrarily distributed real-valued spin systems with bounded support. In our numerical experiments, we demonstrate the effective deterministic training of our proposed models and are able to show interesting features of unsupervised learning which could not be directly observed with sampling. Additionally, we demonstrate how to utilize our TAP-based framework for leveraging trained RBMs as joint priors in denoising problems.
In this paper, based on a successively accuracy-increasing approximation of the $ell_0$ norm, we propose a new algorithm for recovery of sparse vectors from underdetermined measurements. The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the $ell_0$ norm can be controlled. We prove that the series of the approximations asymptotically coincides with the $ell_1$ and $ell_0$ norms when the approximation accuracy changes from the worst fitting to the best fitting. When measurements are noise-free, an optimization scheme is proposed which leads to a number of weighted $ell_1$ minimization programs, whereas, in the presence of noise, we propose two iterative thresholding methods that are computationally appealing. A convergence guarantee for the iterative thresholding method is provided, and, for a particular function in the class of the approximating functions, we derive the closed-form thresholding operator. We further present some theoretical analyses via the restricted isometry, null space, and spherical section properties. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.
Approximate Message Passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problem. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernouilli prior which utilizes a Restricted Boltzmann Machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple iid priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
