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Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager-Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, modes of a Bayesian posterior distribution based upon a Gaussian prior exist and agree with the minimizers of its Onsager-Machlup functional and thus also with weak posterior modes. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric maximum a posteriori estimation is equivalent to Tikhonov-Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of white Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the maximum a posteriori estimator analytically and study its rate of convergence in the small noise limit.

Density-based and classification-based methods have ruled unsupervised anomaly detection in recent years, while reconstruction-based methods are rarely mentioned for the poor reconstruction ability and low performance. However, the latter requires no costly extra training samples for the unsupervised training that is more practical, so this paper focuses on improving this kind of method and proposes a novel Omni-frequency Channel-selection Reconstruction (OCR-GAN) network to handle anomaly detection task in a perspective of frequency. Concretely, we propose a Frequency Decoupling (FD) module to decouple the input image into different frequency components and model the reconstruction process as a combination of parallel omni-frequency image restorations, as we observe a significant difference in the frequency distribution of normal and abnormal images. Given the correlation among multiple frequencies, we further propose a Channel Selection (CS) module that performs frequency interaction among different encoders by adaptively selecting different channels. Abundant experiments demonstrate the effectiveness and superiority of our approach over different kinds of methods, e.g., achieving a new state-of-the-art 98.3 detection AUC on the MVTec AD dataset without extra training data that markedly surpasses the reconstruction-based baseline by +38.1 and the current SOTA method by +0.3. Source code is available at //github.com/zhangzjn/OCR-GAN.

This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction $\tau$ takes as input a length-$n$ binary codeword and injects in an adversarial manner up to $n\tau$ asymmetric errors, i.e., errors that only zero out bits but do not flip $0$'s to $1$'s. It is known that the largest $(L-1)$-list-decodable code for the Z-channel with error fraction $\tau$ has exponential size (in $n$) if $\tau$ is less than a critical value that we call the $(L-1)$-list-decoding Plotkin point and has constant size if $\tau$ is larger than the threshold. The $(L-1)$-list-decoding Plotkin point is known to be $ L^{-\frac{1}{L-1}} - L^{-\frac{L}{L-1}} $, which equals $1/4$ for unique-decoding with $ L-1=1 $. In this paper, we derive various results for the size of the largest codes above and below the list-decoding Plotkin point. In particular, we show that the largest $(L-1)$-list-decodable code $\epsilon$-above the Plotkin point, {for any given sufficiently small positive constant $ \epsilon>0 $,} has size $\Theta_L(\epsilon^{-3/2})$ for any $L-1\ge1$. We also devise upper and lower bounds on the exponential size of codes below the list-decoding Plotkin point.

Multiple systems estimation is a standard approach to quantifying hidden populations where data sources are based on lists of known cases. A typical modelling approach is to fit a Poisson loglinear model to the numbers of cases observed in each possible combination of the lists. It is necessary to decide which interaction parameters to include in the model, and information criterion approaches are often used for model selection. Difficulties in the context of multiple systems estimation may arise due to sparse or nil counts based on the intersection of lists, and care must be taken when information criterion approaches are used for model selection due to issues relating to the existence of estimates and identifiability of the model. Confidence intervals are often reported conditional on the model selected, providing an over-optimistic impression of the accuracy of the estimation. A bootstrap approach is a natural way to account for the model selection procedure. However, because the model selection step has to be carried out for every bootstrap replication, there may be a high or even prohibitive computational burden. We explore the merit of modifying the model selection procedure in the bootstrap to look only among a subset of models, chosen on the basis of their information criterion score on the original data. This provides large computational gains with little apparent effect on inference. Another model selection approach considered and investigated is a downhill search approach among models, possibly with multiple starting points.

Variable selection is a procedure to attain the truly important predictors from inputs. Complex nonlinear dependencies and strong coupling pose great challenges for variable selection in high-dimensional data. In addition, real-world applications have increased demands for interpretability of the selection process. A pragmatic approach should not only attain the most predictive covariates, but also provide ample and easy-to-understand grounds for removing certain covariates. In view of these requirements, this paper puts forward an approach for transparent and nonlinear variable selection. In order to transparently decouple information within the input predictors, a three-step heuristic search is designed, via which the input predictors are grouped into four subsets: the relevant to be selected, and the uninformative, redundant, and conditionally independent to be removed. A nonlinear partial correlation coefficient is introduced to better identify the predictors which have nonlinear functional dependence with the response. The proposed method is model-free and the selected subset can be competent input for commonly used predictive models. Experiments demonstrate the superior performance of the proposed method against the state-of-the-art baselines in terms of prediction accuracy and model interpretability.

We study the online variant of the Min-Sum Set Cover (MSSC) problem, a generalization of the well-known list update problem. In the MSSC problem, an algorithm has to maintain the time-varying permutation of the list of $n$ elements, and serve a sequence of requests $R_1, R_2, \dots, R_t, \dots$. Each $R_t$ is a subset of elements of cardinality at most $r$. For a requested set $R_t$, an online algorithm has to pay the cost equal to the position of the first element from $R_t$ on its list. Then, it may arbitrarily permute its list, paying the number of swapped adjacent element pairs. We present the first constructive deterministic algorithm for this problem, whose competitive ratio does not depend on $n$. Our algorithm is $O(r^2)$-competitive, which beats both the existential upper bound of $O(r^4)$ by Bienkowski and Mucha [AAAI '23] and the previous constructive bound of $O(r^{3/2} \cdot \sqrt{n})$ by Fotakis et al. [ICALP '20]. Furthermore, we show that our algorithm attains an asymptotically optimal competitive ratio of $O(r)$ when compared to the best fixed permutation of elements.

We examine the characteristic activation values of individual ReLU units in neural networks. We refer to the corresponding set for such characteristic activation values in the input space as the characteristic activation set of a ReLU unit. We draw an explicit connection between the characteristic activation set and learned features in ReLU networks. This connection leads to new insights into why various neural network normalization techniques used in modern deep learning architectures regularize and stabilize SGD optimization. Utilizing these insights, we propose a geometric approach to parameterize ReLU networks for improved feature learning. We empirically verify its usefulness with less carefully chosen initialization schemes and larger learning rates. We report improved optimization stability, faster convergence speed, and better generalization performance.

Generative flow networks (GFlowNets) are amortized variational inference algorithms that treat sampling from a distribution over compositional objects as a sequential decision-making problem with a learnable action policy. Unlike other algorithms for hierarchical sampling that optimize a variational bound, GFlowNet algorithms can stably run off-policy, which can be advantageous for discovering modes of the target distribution. Despite this flexibility in the choice of behaviour policy, the optimal way of efficiently selecting trajectories for training has not yet been systematically explored. In this paper, we view the choice of trajectories for training as an active learning problem and approach it using Bayesian techniques inspired by methods for multi-armed bandits. The proposed algorithm, Thompson sampling GFlowNets (TS-GFN), maintains an approximate posterior distribution over policies and samples trajectories from this posterior for training. We show in two domains that TS-GFN yields improved exploration and thus faster convergence to the target distribution than the off-policy exploration strategies used in past work.

Theoretical studies on transfer learning or domain adaptation have so far focused on situations with a known hypothesis class or model; however in practice, some amount of model selection is usually involved, often appearing under the umbrella term of hyperparameter-tuning: for example, one may think of the problem of tuning for the right neural network architecture towards a target task, while leveraging data from a related source task. Now, in addition to the usual tradeoffs on approximation vs estimation errors involved in model selection, this problem brings in a new complexity term, namely, the transfer distance between source and target distributions, which is known to vary with the choice of hypothesis class. We present a first study of this problem, focusing on classification; in particular, the analysis reveals some remarkable phenomena: adaptive rates, i.e., those achievable with no distributional information, can be arbitrarily slower than oracle rates, i.e., when given knowledge on distances.