520 research outputs found
Monotone deep Boltzmann machines
Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods
ever studied, are multi-layered probabilistic models governed by a pairwise
energy function that describes the likelihood of all variables/nodes in the
network. In practice, DBMs are often constrained, i.e., via the
\emph{restricted} Boltzmann machine (RBM) architecture (which does not permit
intra-layer connections), in order to allow for more efficient inference. In
this work, we revisit the generic DBM approach, and ask the question: are there
other possible restrictions to their design that would enable efficient
(approximate) inference? In particular, we develop a new class of restricted
model, the monotone DBM, which allows for arbitrary self-connection in each
layer, but restricts the \emph{weights} in a manner that guarantees the
existence and global uniqueness of a mean-field fixed point. To do this, we
leverage tools from the recently-proposed monotone Deep Equilibrium model and
show that a particular choice of activation results in a fixed-point iteration
that gives a variational mean-field solution. While this approach is still
largely conceptual, it is the first architecture that allows for efficient
approximate inference in fully-general weight structures for DBMs. We apply
this approach to simple deep convolutional Boltzmann architectures and
demonstrate that it allows for tasks such as the joint completion and
classification of images, within a single deep probabilistic setting, while
avoiding the pitfalls of mean-field inference in traditional RBMs
Analysis of the accuracy and convergence of equation-free projection to a slow manifold
In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a
Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl.
Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms
within the context of equation-free methods to approximate low-dimensional,
attracting, slow manifolds in systems of differential equations with multiple
time scales. For user-specified values of a finite number of the observables,
the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an
approximation of the appropriate zero of the (m+1)-st time derivative of the
remaining variables and uses this root to approximate the location of the point
on the slow manifold corresponding to these values of the observables. This
article is the first of two articles in which the accuracy and convergence of
the iterative algorithms are analyzed. Here, we work directly with explicit
fast--slow systems, in which there is an explicit small parameter, epsilon,
measuring the separation of time scales. We show that, for each m = 0, 1, ...,
the fixed point of the iterative algorithm approximates the slow manifold up to
and including terms of O(epsilon^m). Moreover, for each m, we identify
explicitly the conditions under which the m-th iterative algorithm converges to
this fixed point. Finally, we show that when the iteration is unstable (or
converges slowly) it may be stabilized (or its convergence may be accelerated)
by application of the Recursive Projection Method. Alternatively, the
Newton-Krylov Generalized Minimal Residual Method may be used. In the
subsequent article, we will consider the accuracy and convergence of the
iterative algorithms for a broader class of systems-in which there need not be
an explicit small parameter-to which the algorithms also apply
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