A Connection Between GRBF and MLP

Both multilayer perceptrons (MLP) and Generalized Radial Basis Functions (GRBF) have good approximation properties, theoretically and experimentally. Are they related? The main point of this paper is to show that for normalized inputs, multilayer perceptron networks are radial function networks (albeit with a non-standard radial function). This provides an interpretation of the weights w as centers t of the radial function network, and therefore as equivalent to templates. This insight may be useful for practical applications, including better initialization procedures for MLP. In the remainder of the paper, we discuss the relation between the radial functions that correspond to the sigmoid for normalized inputs and well-behaved radial basis functions, such as the Gaussian. In particular, we observe that the radial function associated with the sigmoid is an activation function that is good approximation to Gaussian basis functions for a range of values of the bias parameter. The implication is that a MLP network can always simulate a Gaussian GRBF network (with the same number of units but less parameters); the converse is true only for certain values of the bias parameter. Numerical experiments indicate that this constraint is not always satisfied in practice by MLP networks trained with backpropagation. Multiscale GRBF networks, on the other hand, can approximate MLP networks with a similar number of parameters.

Fast Pose Estimation with Parameter Sensitive Hashing

Example-based methods are effective for parameter estimation problems when the underlying system is simple or the dimensionality of the input is low. For complex and high-dimensional problems such as pose estimation, the number of required examples and the computational complexity rapidly becme prohibitively high. We introduce a new algorithm that learns a set of hashing functions that efficiently index examples relevant to a particular estimation task. Our algorithm extends a recently developed method for locality-sensitive hashing, which finds approximate neighbors in time sublinear in the number of examples. This method depends critically on the choice of hash functions; we show how to find the set of hash functions that are optimally relevant to a particular estimation problem. Experiments demonstrate that the resulting algorithm, which we call Parameter-Sensitive Hashing, can rapidly and accurately estimate the articulated pose of human figures from a large database of example images.

Parameter Estimation in Chaotic Systems

This report examines how to estimate the parameters of a chaotic system given noisy observations of the state behavior of the system. Investigating parameter estimation for chaotic systems is interesting because of possible applications for high-precision measurement and for use in other signal processing, communication, and control applications involving chaotic systems. In this report, we examine theoretical issues regarding parameter estimation in chaotic systems and develop an efficient algorithm to perform parameter estimation. We discover two properties that are helpful for performing parameter estimation on non-structurally stable systems. First, it turns out that most data in a time series of state observations contribute very little information about the underlying parameters of a system, while a few sections of data may be extraordinarily sensitive to parameter changes. Second, for one-parameter families of systems, we demonstrate that there is often a preferred direction in parameter space governing how easily trajectories of one system can "shadow'" trajectories of nearby systems. This asymmetry of shadowing behavior in parameter space is proved for certain families of maps of the interval. Numerical evidence indicates that similar results may be true for a wide variety of other systems. Using the two properties cited above, we devise an algorithm for performing parameter estimation. Standard parameter estimation techniques such as the extended Kalman filter perform poorly on chaotic systems because of divergence problems. The proposed algorithm achieves accuracies several orders of magnitude better than the Kalman filter and has good convergence properties for large data sets.

Policy Improvement for POMDPs Using Normalized Importance Sampling

We present a new method for estimating the expected return of a POMDP from experience. The estimator does not assume any knowle ge of the POMDP and allows the experience to be gathered with an arbitrary set of policies. The return is estimated for any new policy of the POMDP. We motivate the estimator from function-approximation and importance sampling points-of-view and derive its theoretical properties. Although the estimator is biased, it has low variance and the bias is often irrelevant when the estimator is used for pair-wise comparisons.We conclude by extending the estimator to policies with memory and compare its performance in a greedy search algorithm to the REINFORCE algorithm showing an order of magnitude reduction in the number of trials required.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.