Normalizing flow-based variational inference (flow VI) is a promising approximate inference approach, but its performance remains inconsistent across studies. Numerous algorithmic choices influence flow VI’s performance. We conduct a step-by-step analysis to disentangle the impact of some of the key factors: capacity, objectives, gradient estimators, number of gradient estimates (batchsize), and step-sizes. Each step examines one factor while neutralizing others using insights from the previous steps and/or using extensive parallel computation. To facilitate high-fidelity evaluation, we curate a benchmark of synthetic targets that represent common posterior pathologies and allow for exact sampling. We provide specific recommendations for different factors and propose a flow VI recipe that matches or surpasses leading turnkey Hamiltonian Monte Carlo (HMC) methods.

Predictive posterior densities (PPDs) are essential in approximate inference for quantifying predictive uncertainty and comparing inference methods. Typically, PPDs are estimated by simple Monte Carlo (MC) averages. In this paper, we expose a critical under-recognized issue: the signal-to-noise ratio (SNR) of the simple MC estimator can sometimes be extremely low, leading to unreliable estimates. Our main contribution is a theoretical analysis demonstrating that even with exact inference, SNR can decay rapidly with an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of test data relative to training data. Through several examples, we empirically verify these claims and show that these factors indeed lead to poor SNR and unreliable PPD estimates (sometimes, estimates are off by hundreds of nats even with a million samples). While not the primary focus, we also explore an adaptive importance sampling approach as an illustrative way to mitigate the problem, where we learn the proposal distribution by maximizing a variational proxy to the SNR. Taken together, our findings highlight an important challenge and provide essential insights for reliable estimation.

Many domain experts do not have the time or expertise to write formal Bayesian models. This paper takes an informal problem description as input, and combines a large language model and a probabilistic programming language to define a joint distribution over formal models, latent variables, and data. A posterior over latent variables follows by conditioning on observed data and integrating over formal models. This presents a challenging inference problem. We suggest an inference recipe that amounts to generating many formal models from the large language model, performing approximate inference on each, and then doing a weighted aver- age. This is justified and analyzed as a combination of self-normalized importance sampling, MCMC, and importance-weighted variational inference. Experimentally, this produces sensible predictions from only data and an informal problem description, without the need to specify a formal model.

Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow‐based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model’s prior. Guided by this theoretical insight, we introduce the Model‐Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state‐of‐the‐art performance across a suite of hierarchical and non‐hierarchical benchmarks.

Black-box variational inference performance is sometimes hindered by the use of gradient estimators with high variance. This variance comes from two sources of randomness: Data subsampling and Monte Carlo sampling. While existing control variates only address Monte Carlo noise, and incremental gradient methods typically only address data subsampling, we propose a new "joint" control variate that jointly reduces variance from both sources of noise. This significantly reduces gradient variance, leading to faster optimization in several applications.

Simulation-based inference has been popular for amortized Bayesian computation. It is typical to have more than one posterior approximation, from different inference algorithms, different architectures, or simply the randomness of initialization and stochastic gradients. With a consistency guarantee, we present a general posterior stacking framework to make use of all available approximations. Our stacking method is able to combine densities, simulation draws, confidence intervals, and moments, and address the overall precision, calibration, coverage, and bias of the posterior approximation at the same time. We illustrate our method on several benchmark simulations and a challenging cosmological inference task.

Bayesian reasoning in linear mixed-effects models (LMMs) is challenging and often requires advanced sampling techniques like Markov chain Monte Carlo (MCMC). A common approach is to write the model in a probabilistic programming language and then sample via Hamiltonian Monte Carlo (HMC). However, there are many ways a user can transform a model that make inference more or less efficient. In particular, marginalizing some variables can greatly improve inference but is difficult for users to do manually. We develop an algorithm to easily marginalize random effects in LMMs. A naive approach introduces cubic time operations within an inference algorithm like HMC, but we reduce the running time to linear using fast linear algebra techniques. We show that marginalization is always beneficial when applicable and highlight improvements in various models, especially ones from cognitive sciences.

Black-box variational inference (BBVI) is a general-purpose approximate inference approach that converts inference to a stochastic optimization problem. However, the difficulty of solving the BBVI optimization problem reliably and robustly using stochastic gradient methods has limited its applicability. We present a novel optimization approach for BBVI using the sample average approximation (SAA). SAA converts stochastic problems to deterministic ones by optimizing over a fixed random sample, which enables optimization tools such as quasi-Newton methods and line search that bypass the difficulties faced by stochastic gradient methods. We design an approach called "SAA for VI" that solves a sequence of SAA problems with increasing sample sizes to reliably and robustly solve BBVI problems without problem-specific tuning. We focus on quasi-Newton methods, which are well suited to problems with up to hundreds of latent variables. Our experiments show that SAA for VI simplifies the VI problem and achieves faster performance than existing methods.

Many methods that build powerful variational distributions based on unadjusted Langevin transitions exist. Most of these were developed using a wide range of different approaches and techniques. Unfortunately, the lack of a unified analysis and derivation makes developing new methods and reasoning about existing ones a challenging task. We address this giving a single analysis that unifies and generalizes these existing techniques. The main idea is to augment the target and variational by numerically simulating the underdamped Langevin diffusion process and its time reversal. The benefits of this approach are twofold: it provides a unified formulation for many existing methods, and it simplifies the development of new ones. In fact, using our formulation we propose a new method that combines the strengths of previously existing algorithms; it uses underdamped Langevin transitions and powerful augmentations parameterized by a score network. Our empirical evaluation shows that our proposed method consistently outperforms relevant baselines in a wide range of tasks.

Black-box variational inference is widely used in situations where there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs—namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides rigorous guarantees that methods similar to those used in practice converge on realistic inference problems.

To check the accuracy of Bayesian computations, it is common to use rank-based simulation-based calibration (SBC). However, SBC has drawbacks: The test statistic is somewhat ad-hoc, interactions are difficult to examine, multiple testing is a challenge, and the resulting p-value is not a divergence metric. We propose to replace the marginal rank test with a flexible classification approach that learns test statistics from data. This measure typically has a higher statistical power than the SBC test and returns an interpretable divergence measure of miscalibration, computed from classification accuracy. This approach can be used with different data generating processes to address simulation-based inference or traditional inference methods like Markov chain Monte Carlo or variational inference. We illustrate an automated implementation using neural networks and statistically-inspired features, and validate the method with numerical and real data experiments.

Variational inference for state space models (SSMs) is known to be hard in general. Recent works focus on deriving variational objectives for SSMs from unbiased sequential Monte Carlo estimators. We reveal that the marginal particle filter is obtained from sequential Monte Carlo by applying Rao-Blackwellization operations, which sacrifices the trajectory information for reduced variance and differentiability. We propose the variational marginal particle filter (VMPF), which is a differentiable and reparameterizable variational filtering objective for SSMs based on an unbiased estimator. We find that VMPF with biased gradients gives tighter bounds than previous objectives, and the unbiased reparameterization gradients are sometimes beneficial.

Hierarchical models represent a challenging setting for inference algorithms. MCMC methods struggle to scale to large models with many local variables and observations, and variational inference (VI) may fail to provide accurate approximations due to the use of simple variational families. Some variational methods (e.g. importance weighted VI) integrate Monte Carlo methods to give better accuracy, but these tend to be unsuitable for hierarchical models, as they do not allow for subsampling and their performance tends to degrade for high dimensional models. We propose a new family of variational bounds for hierarchical models, based on the application of tightening methods (e.g. importance weighting) separately for each group of local random variables. We show that our approach naturally allows the use of subsampling to get unbiased gradients, and that it fully leverages the power of methods that build tighter lower bounds by applying them independently in lower dimensional spaces, leading to better results and more accurate posterior approximations than relevant baselines.

Several approximate inference algorithms have been proposed to minimize an alpha-divergence between an approximating distribution and a target distribution. Many of these algorithms introduce bias, the magnitude of which becomes problematic in high dimensions. Other algorithms are unbiased. These often seem to suffer from high variance, but little is rigorously known. In this work we study unbiased methods for alpha-divergence minimization through the Signal-to-Noise Ratio (SNR) of the gradient estimator. We study several representative scenarios where strong analytical results are possible, such as fully-factorized or Gaussian distributions. We find that when alpha is not zero, the SNR worsens exponentially in the dimensionality of the problem. This casts doubt on the practicality of these methods. We empirically confirm these theoretical results.

It is difficult to use subsampling with variational inference in hierarchical models since the number of local latent variables scales with the dataset. Thus, inference in hierarchical models remains a challenge at a large scale. It is helpful to use a variational family with a structure matching the posterior, but optimization is still slow due to the huge number of local distributions. Instead, this paper suggests an amortized approach where shared parameters simultaneously represent all local distributions. This approach is similarly accurate as using a given joint distribution (e.g., a full-rank Gaussian) but is feasible on datasets that are several orders of magnitude larger. It is also dramatically faster than using a structured variational distribution.

Given an unnormalized target distribution we want to obtain approximate samples from it and a tight lower bound on its (log) normalization constant log Z. Annealed Importance Sampling (AIS) with Hamiltonian MCMC is a powerful method that can be used to do this. Its main drawback is that it uses non-differentiable transition kernels, which makes tuning its many parameters hard. We propose a framework to use an AIS-like procedure with Uncorrected Hamiltonian MCMC, called Uncorrected Hamiltonian Annealing. Our method leads to tight and differentiable lower bounds on log Z. We show empirically that our method yields better performances than other competing approaches, and that the ability to tune its parameters using reparameterization gradients may lead to large performance improvements.

Stochastic gradient descent (SGD) is the workhorse of modern machine learning. Sometimes, there are many different potential gradient estimators that can be used. When so, choosing the one with the best tradeoff between cost and variance is important. This paper analyzes the convergence rates of SGD as a function of time, rather than iterations. This results in a simple rule to select the estimator that leads to the best optimization convergence guarantee. This choice is the same for different variants of SGD, and with different assumptions about the objective (e.g. convexity or smoothness). Inspired by this principle, we propose a technique to automatically select an estimator when a finite pool of estimators is given. Then, we extend to infinite pools of estimators, where each one is indexed by control variate weights. Empirically, automatically choosing an estimator performs comparably to the best estimator chosen with hindsight.

Black-box variational inference tries to approximate a complex target distribution through a gradient-based optimization of the parameters of a simpler distribution. Provable convergence guarantees require structural properties of the objective. This paper shows that for location-scale family approximations, if the target is M-Lipschitz smooth, then so is the “energy” part of the variational objective. The key proof idea is to describe gradients in a certain inner-product space, thus permitting the use of Bessel’s inequality. This result gives bounds on the location of the optimal parameters, and is a key ingredient for convergence guarantees.

Recent research has seen several advances relevant to black-box VI, but the current state of automatic posterior inference is unclear. One such advance is the use of normalizing flows to define flexible posterior densities for deep latent variable models. Another direction is the integration of Monte-Carlo methods to serve two purposes; first, to obtain tighter variational objectives for optimization, and second, to define enriched variational families through sampling. However, both flows and variational Monte-Carlo methods remain relatively unexplored for black-box VI. Moreover, on a pragmatic front, there are several optimization considerations like step-size scheme, parameter initialization, and choice of gradient estimators, for which there are no clear guidance in the existing literature. In this paper, we postulate that black-box VI is best addressed through a careful combination of numerous algorithmic components. We evaluate components relating to optimization, flows, and Monte-Carlo methods on a benchmark of 30 models from the Stan model library. The combination of these algorithmic components significantly advances the state-of-the-art "out of the box" variational inference.

Flexible variational distributions improve variational inference but are harder to optimize. In this work we present a control variate that is applicable for any reparameterizable distribution with known mean and covariance, e.g. Gaussians with any covariance structure. The control variate is based on a quadratic approximation of the model, and its parameters are set using a double-descent scheme. We empirically show that this control variate leads to large improvements in gradient variance and optimization convergence for inference with non-factorized variational distributions.

Recent variational inference methods use stochastic gradient estimators whose variance is not well understood. Theoretical guarantees for these estimators are important to understand when these methods will or will not work. This paper gives bounds for the common “reparameterization” estimators when the target is smooth and the variational family is a location-scale distribution. These bounds are unimprovable and thus provide the best possible guarantees under the stated assumptions.

Recent work in variational inference (VI) has used ideas from Monte Carlo estimation to obtain tighter lower bounds on the log-likelihood to be used as objectives for VI. However, there is not a systematic understanding of how optimizing different objectives relates to approximating the posterior distribution. Developing such a connection is important if the ideas are to be applied to inference—i.e., applications that require an approximate posterior and not just an approximation of the log-likelihood. Given a VI objective defined by a Monte Carlo estimator of the likelihood, we use a "divide and couple" procedure to identify augmented proposal and target distributions so that the gap between the VI objective and the log-likelihood is equal to the divergence between these distributions. Thus, after maximizing the VI objective, the augmented variational distribution may be used to approximate the posterior distribution.

We study the effects of approximate inference on the performance of Thompson sampling in the $k$-armed bandit problems. Thompson sampling is a successful algorithm for online decision-making but requires posterior inference, which often must be approximated in practice. We show that even small constant inference error (in $\alpha$-divergence) can lead to poor performance (linear regret) due to under-exploration (for $\alpha<1$) or over-exploration (for $\alpha>0$) by the approximation. While for $\alpha > 0$ this is unavoidable, for $\alpha \leq 0$ the regret can be improved by adding a small amount of forced exploration even when the inference error is a large constant.

Recent work used importance sampling ideas for better variational bounds on likelihoods. We clarify the applicability of these ideas to pure probabilistic inference, by showing the resulting Importance Weighted Variational Inference (IWVI) technique is an instance of augmented variational inference, thus identifying the looseness in previous work. Experiments confirm IWVI's practicality for probabilistic inference. As a second contribution, we investigate inference with elliptical distributions, which improves accuracy in low dimensions, and convergence in high dimensions.

Variational inference is increasingly being addressed with stochastic optimization. In this setting, the gradient's variance plays a crucial role in the optimization procedure, since high variance gradients lead to poor convergence. A popular approach used to reduce gradient's variance involves the use of control variates. Despite the good results obtained, control variates developed for variational inference are typically looked at in isolation. In this paper we clarify the large number of control variates that are available by giving a systematic view of how they are derived. We also present a Bayesian risk minimization framework in which the quality of a procedure for combining control variates is quantified by its effect on optimization convergence rates, which leads to a very simple combination rule. Results show that combining a large number of control variates this way significantly improves the convergence of inference over using the typical gradient estimators or a reduced number of control variates.

Two popular classes of methods for approximate inference are Markov chain Monte Carlo (MCMC) and variational inference. MCMC tends to be accurate if run for a long enough time, while variational inference tends to give better approximations at shorter time horizons. However, the amount of time needed for MCMC to exceed the performance of variational methods can be quite high, motivating more fine-grained tradeoffs. This paper derives a distribution over variational parameters, designed to minimize a bound on the divergence between the resulting marginal distribution and the target, and gives an example of how to sample from this distribution in a way that interpolates between the behavior of existing methods based on Langevin dynamics and stochastic gradient variational inference (SGVI).

We examine the effect of clamping variables for approximate inference in undirected graphical models with pairwise relationships and discrete variables. For any number of variable labels, we demonstrate that clamping and summing approximate sub-partition functions can lead only to a decrease in the partition function estimate for TRW, and an increase for the naive mean field method, in each case guaranteeing an improvement in the approximation and bound. We next focus on binary variables, add the Bethe approximation to consideration and examine ways to choose good variables to clamp, introducing new methods. We show the importance of identifying highly frustrated cycles, and of checking the singleton entropy of a variable. We explore the value of our methods by empirical analysis and draw lessons to guide practitioners.

Bayesian decision-theory underpins robust decision-making in applications ranging from plant control to robotics where hedging action selection against state uncertainty is critical for minimizing low probability but potentially catastrophic outcomes (e.g, uncontrollable plant conditions or robots falling into stairwells). Unfortunately, belief state distributions in such settings are often complex and/or high dimensional, thus prohibiting the efficient application of analytical techniques for expected utility computation when real-time control is required. This leaves Monte Carlo evaluation as one of the few viable (and hence frequently used) techniques for online action selection. However, loss-insensitive Monte Carlo methods may require large numbers of samples to identify optimal actions with high certainty since they may sample from highprobability regions that do not disambiguate action utilities. In this paper we remedy this problem by deriving an optimal proposal distribution for a loss-calibrated Monte Carlo importance sampler that bounds the regret of using an estimated optimal action. Empirically, we show that using our loss-calibrated Monte Carlo method yields high-accuracy optimal action selections in a fraction of the number of samples required by conventional loss-insensitive samplers.

Hamiltonian Monte Carlo (HMC) is a successful approach for sampling from continuous densities. However, it has difficulty simulating Hamiltonian dynamics with non-smooth functions, leading to poor performance. This paper is motivated by the behavior of Hamiltonian dynamics in physical systems like optics. We introduce a modification of the Leapfrog discretization of Hamiltonian dynamics on piecewise continuous energies, where intersections of the trajectory with discontinuities are detected, and the momentum is reflected or refracted to compensate for the change in energy. We prove that this method preserves the correct stationary distribution when boundaries are affine. Experiments show that by reducing the number of rejected samples, this method improves on traditional HMC.

Inference is typically intractable in high-treewidth undirected graphical models, making maximum likelihood learning a challenge. One way to overcome this is to restrict parameters to a tractable set, most typically the set of tree-structured parameters. This paper explores an alternative notion of a tractable set, namely a set of “fast-mixing parameters” where Markov chain Monte Carlo (MCMC) inference can be guaranteed to quickly converge to the stationary distribution. While it is common in practice to approximate the likelihood gradient using samples obtained from MCMC, such procedures lack theoretical guarantees. This paper proves that for any exponential family with bounded sufficient statistics, (not just graphical models) when parameters are constrained to a fast-mixing set, gradient descent with gradients approximated by sampling will approximate the maximum likelihood solution inside the set with high-probability. When unregularized, to find a solution epsilon-accurate in log-likelihood requires a total amount of effort cubic in 1/epsilon, disregarding logarithmic factors. When ridge-regularized, strong convexity allows a solution epsilon-accurate in parameter distance with an effort quadratic in 1/epsilon. Both of these provide of a fully-polynomial time randomized approximation scheme.

Recent advances in optimization theory have shown that smooth strongly convex finite sums can be minimized faster than by treating them as a black box "batch" problem. In this work we introduce a new method in this class with a theoretical convergence rate four times faster than existing methods, for sums with sufficiently many terms. This method is also amendable to a sampling without replacement scheme that in practice gives further speed-ups. We give empirical results showing state of the art performance.

A successful approach to structured learning is to write the learning objective as a joint function of linear parameters and inference messages, and iterate between updates to each. This paper observes that if the inference problem is “smoothed” through the addition of entropy terms, for fixed messages, the learning objective reduces to a traditional (non-structured) logistic regression problem with respect to parameters. In these logistic regression problems, each training example has a bias term determined by the current set of messages. Based on this insight, the structured energy function can be extended from linear factors to any function class where an “oracle” exists to minimize a logistic loss.

Inference in general Ising models is difficult, due to high treewidth making tree-based algorithms intractable. Moreover, when interactions are strong, Gibbs sampling may take exponential time to converge to the stationary distribution. We present an algorithm to project Ising model parameters onto a parameter set that is guaranteed to be fast mixing, under several divergences. We find that Gibbs sampling using the projected parameters is more accurate than with the original parameters when interaction strengths are strong and when limited time is available for sampling.

Likelihood-based learning of graphical models faces challenges of computational complexity and robustness to model misspecification. This paper studies methods that fit parameters directly to maximize a measure of the accuracy of predicted marginals, taking into account both model and inference approximations at training time. Experiments on imaging problems suggest marginalization-based learning performs better than likelihood-based approximations on difficult problems where the model being fit is approximate in nature.

We present a dual decomposition approach to the tree-reweighted belief propagation objective. Each tree in the tree-reweighted bound yields one subproblem, which can be solved with the sum-product algorithm. The master problem is a simple differentiable optimization, to which a standard optimization method can be applied. Experimental results on 10x10 Ising models show the dual decomposition approach using L-BFGS is similar in settings where message-passing converges quickly, and one to two orders of magnitude faster in settings where message-passing requires many iterations, specifically high accuracy convergence, and strong interactions.

Training of conditional random fields often takes the form of a double-loop procedure with message-passing inference in the inner loop. This can be very expensive, as the need to solve the inner loop to high accuracy can require many message-passing iterations. This paper seeks to reduce the expense of such training, by redefining the training objective in terms of the approximate marginals obtained after message-passing is “truncated” to a fixed number of iterations. An algorithm is derived to efficiently compute the exact gradient of this objective. On a common pixel labeling benchmark, this procedure improves training speeds by an order of magnitude, and slightly improves inference accuracy if a very small number of message-passing iterations are used at test time.

This paper proposes a simple and efficient finite difference method for implicit differentiation of marginal inference results in discrete graphical models. Given an arbitrary loss function, defined on marginals, we show that the derivatives of this loss with respect to model parameters can be obtained by running the inference procedure twice, on slightly perturbed model parameters. This method can be used with approximate inference, with a loss function over approximate marginals. Convenient choices of loss functions make it practical to fit graphical models with hidden variables, high treewidth and/or model misspecification.

Since cameras blur the incoming light during measurement, different images of the same surface do not contain the same information about that surface. Thus, in general, corresponding points in multiple views of a scene have different image intensities. While multiple-view geometry constrains the locations of corresponding points, it does not give relationships between the signals at corresponding locations. This paper offers an elementary treatment of these relationships. We first develop the notion of "ideal" and "real" images, corresponding to, respectively, the raw incoming light and the measured signal. This framework separates the filtering and geometric aspects of imaging. We then consider how to synthesize one view of a surface from another; if the transformation between the two views is affine, it emerges that this is possible if and only if the singular values of the affine matrix are positive. Next, we consider how to combine the information in several views of a surface into a single output image. By developing a new tool called "frequency segmentation," we show how this can be done despite not knowing the blurring kernel.

Prior distributions are useful for robust low-level vision, and undirected models (e.g. Markov Random Fields) have become a central tool for this purpose. Though sometimes these priors can be specified by hand, this becomes difficult in large models, which has motivated learning these models from data. However, maximum likelihood learning of undirected models is extremely difficult- essentially all known methods require approximations and/or high computational cost. Conversely, directed models are essentially trivial to learn from data, but have not received much attention for low-level vision. We compare the two formalisms of directed and undirected models, and conclude that there is no a priori reason to believe one better represents low-level vision quantities. We formulate two simple directed priors, for natural images and stereo disparity, to empirically test if the undirected formalism is superior. We find in both cases that a simple directed model can achieve results similar to the best learnt undirected models with significant speedups in training time, suggesting that directed models are an attractive choice for tractable learning.

We analytically develop a filter that is able to measure the linear stretch of the transformation around a point, and present results of applying it to real signals. We show that this method is a real-time alternative solution for measuring local signal transformations. Experimentally, this method can accurately measure stretch, however, it is sensitive to shift.

This paper presents a new constraint connecting the signals in multiple views of a surface. The constraint arises from a harmonic analysis of the geometry of the imaging process and it gives rise to a new technique for multiple view image reconstruction. Given several views of a surface from different positions, fundamentally different information is present in each image, owing to the fact that cameras measure the incoming light only after the application of a low-pass filter. Our analysis shows how the geometry of the imaging is connected to this filtering. This leads to a technique for constructing a single output image containing all the information present in the input images.

This paper proposes the "epipolar pencil transformation" (EPT). This is a tool for comparing the signals in different images, with no use of feature detection, yet taking advantage of the constraints given by epipolar geometry. The idea is to develop a descriptor for each point, summarizing the signals on the pencil of lines intersecting at that point. To compute the EPT, first find compact descriptors for each line, then combine these appropriately for each pencil. Given the EPT for two images, computing the epipolar geometry reduces to a closest pairs problem- select one pencil from each set such that the L1distance (in descriptor space) is minimized. By this reduction to a high dimensional closest pairs problem, recent advances in computational geometry can be used to efficiently identify the best global solution. This technique is robust, as each potential solution is evaluated by comparing the signals for all the lines passing through the two hypothesized epipoles. At the same time, as the closest pairs algorithm performs a global search, the solution is not distracted by local minima. The EPT is used here both for the problem of two-view rigid motion, and many-view place recognition.

We develop a theoretical basis for creating color histograms that are in-variant under deformation or changes in viewpoint. The gradients in differ-ent color channels weight the influence of a pixel on the histogram so as to cancel out the changes induced by deformations. Experiments show these histograms to be invariant under a variety of distortions and changes in view-point. 1

We present a probabilistic framework for visual correspondence, inertial measurements and egomotion. First, we describe a simple method based on Gabor filters to produce correspondence probability distributions. Next, we generate a noise model for inertial measurements. Probability distributions over the motions are then computed directly from the correspondence distributions and the inertial measurements. We investigate combining the inertial and visual information for a single distribution over the motions. We find that with smaller amounts of correspondence information, fusion of the visual data with the inertial sensor results in much better egomotion estimation. This is essentially because inertial measurements decrease the "translation-rotation" ambiguity. However, when more correspondence information is used, this ambiguity is reduced to such a degree that the inertial measurements provide negligible improvement in accuracy. This suggests that inertial and visual information are more closely integrated in a compositional sense