The Hamiltonian Monte Carlo (HMC) algorithm is often lauded for its ability to effectively sample from high-dimensional distributions. In this paper we challenge the presumed domination of HMC for the Bayesian analysis of GLMs. By utilizing the structure of the compute graph rather than the graphical model, we show a reduction of the time per sweep of a full-scan Gibbs sampler from $O(d^2)$ to $O(d)$, where $d$ is the number of GLM parameters. A simple change to the implementation of the Gibbs sampler allow us to perform Bayesian inference on high-dimensional GLMs that are practically infeasible with traditional Gibbs sampler implementations. We empirically demonstrate a substantial increase in effective sample size per time when comparing our Gibbs algorithms to state-of-the-art HMC algorithms. While Gibbs is superior in terms of dimension scaling, neither Gibbs nor HMC dominate the other: we provide numerical and theoretical evidence that HMC retains an edge in certain circumstances thanks to its advantageous condition number scaling. Interestingly, for GLMs of fixed data size, we observe that increasing dimensionality can stabilize or even decrease condition number, shedding light on the empirical advantage of our efficient Gibbs sampler.

The performance of sequential Monte Carlo (SMC) samplers heavily depends on the tuning of the Markov kernels used in the path proposal. For SMC samplers with unadjusted Markov kernels, standard tuning objectives, such as the Metropolis-Hastings acceptance rate or the expected-squared jump distance, are no longer applicable. While stochastic gradient-based end-to-end optimization algorithms have been explored for tuning SMC samplers, they often incur excessive training costs, even for tuning just the kernel step sizes. In this work, we propose a general adaptation framework for tuning the Markov kernels in SMC samplers by minimizing the incremental Kullback-Leibler (KL) divergence between the proposal and target paths. For step size tuning, we provide a gradient- and tuning-free algorithm that is generally applicable for kernels such as Langevin Monte Carlo (LMC). We further demonstrate the utility of our approach by providing a tailored scheme for tuning kinetic LMC used in SMC samplers. Our implementations are able to obtain a full schedule of tuned parameters at the cost of a few vanilla SMC runs, which is a fraction of gradient-based approaches.

Most expressive variational families---such as normalizing flows---lack practical convergence guarantees, as their theoretical assurances typically hold only at the intractable global optimum. In this work, we present a general recipe for constructing tuning-free, asymptotically exact variational flows on *arbitrary* state spaces from involutive MCMC kernels. The core methodological component is a novel representation of general involutive MCMC kernels as invertible, measure- preserving iterated random function systems, which act as the flow maps of our variational flows. This leads to three new variational families with provable total variation convergence. Our framework resolves key practical limitations of existing variational families with similar guarantees (e.g., MixFlows), while requiring substantially weaker theoretical assumptions. Finally, we demonstrate the competitive performance of our flows across tasks including posterior approximation, Monte Carlo estimates, and normalization constant estimation, outperforming or matching No-U-Turn sampler (NUTS) and black-box normalizing flows.

Continuous Conditional Generative Adversarial Networks (CcGANs) enable generative modeling conditional on continuous scalar variables (termed regression labels). However, they can produce subpar fake images due to limited training data. Although Negative Data Augmentation (NDA) effectively enhances unconditional and class-conditional GANs by introducing anomalies into real training images, guiding the GANs away from low-quality outputs, its impact on CcGANs is limited, as it fails to replicate negative samples that may occur during the CcGAN sampling. We present a novel NDA approach called Dual-NDA specifically tailored for CcGANs to address this problem. Dual-NDA employs two types of negative samples: visually unrealistic images generated from a pre-trained CcGAN and label-inconsistent images created by manipulating real images' labels. Leveraging these negative samples, we introduce a novel discriminator objective alongside a modified CcGAN training algorithm. Empirical analysis on UTKFace and Steering Angle reveals that Dual-NDA consistently enhances the visual fidelity and label consistency of fake images generated by CcGANs, exhibiting a substantial performance gain over the vanilla NDA. Moreover, by applying Dual-NDA, CcGANs demonstrate a remarkable advancement beyond the capabilities of state-of-the-art conditional GANs and diffusion models, establishing a new pinnacle of performance. Our codes can be found at https://github.com/UBCDingXin/Dual-NDA.

Multimodal representation learning techniques typically require paired samples to learn shared representations, but collecting paired samples can be challenging in fields like biology, where measurement devices often destroy the samples. This paper presents an approach to address the challenge of aligning unpaired samples across disparate modalities in multimodal representation learning. We draw an analogy between potential outcomes in causal inference and potential views in multimodal observations, allowing us to leverage Rubin's framework to estimate a common space for matching samples. Our approach assumes experimentally perturbed samples by treatments, and uses this to estimate a propensity score from each modality. We show that the propensity score encapsulates all shared information between a latent state and treatment, and can be used to define a distance between samples. We experiment with two alignment techniques that leverage this distance---shared nearest neighbours (SNN) and optimal transport (OT) matching---and find that OT matching results in significant improvements over state-of-the-art alignment approaches in on synthetic multi-modal tasks, in real-world data from NeurIPS Multimodal Single-Cell Integration Challenge, and on a single cell microscopy to expression prediction task.

This work presents mixed variational flows (MixFlows), a new variational family that consists of a mixture of repeated applications of a map to an initial reference distribution. First, we provide efficient algorithms for i.i.d. sampling, density evaluation, and unbiased ELBO estimation. We then show that MixFlows have MCMC-like convergence guarantees when the flow map is ergodic and measure-preserving, and provide bounds on the accumulation of error for practical implementations where the flow map is approximated. Finally, we develop an implementation of MixFlows based on uncorrected discretized Hamiltonian dynamics combined with deterministic momentum refreshment. Simulated and real data experiments show that MixFlows can provide more reliable posterior approximations than several black-box normalizing flows, as well as samples of comparable quality to those obtained from state-of-the-art MCMC methods.

In this paper, we investigate the impact of numerical instability on the reliability of sampling, density evaluation, and evidence lower bound (ELBO) estimation in variational flows. We first empirically demonstrate that common flows can exhibit a catastrophic accumulation of error: the numerical flow map deviates significantly from the exact map---which affects sampling---and the numerical inverse flow map does not accurately recover the initial input---which affects density and ELBO computations. Surprisingly though, we find that results produced by flows are often accurate enough for applications despite the presence of serious numerical instability. In this work, we treat variational flows as chaotic dynamical systems, and leverage shadowing theory to elucidate this behavior via theoretical guarantees on the error of sampling, density evaluation, and ELBO estimation. Finally, we develop and empirically test a diagnostic procedure that can be used to validate results produced by numerically unstable flows in practice.

This article focuses on conditional generative modeling (CGM) for image data with continuous, scalar conditions (termed regression labels). We propose the first model for this task which is called continuous conditional generative adversarial network (CcGAN). Existing conditional GANs (cGANs) are mainly designed for categorical conditions (e.g., class labels). Conditioning on regression labels is mathematically distinct and raises two fundamental problems: (P1) since there may be very few (even zero) real images for some regression labels, minimizing existing empirical versions of cGAN losses (a.k.a. empirical cGAN losses) often fails in practice; and (P2) since regression labels are scalar and infinitely many, conventional label input mechanisms (e.g., combining a hidden map of the generator/discriminator with a one-hot encoded label) are not applicable. We solve these problems by: (S1) reformulating existing empirical cGAN losses to be appropriate for the continuous scenario; and (S2) proposing a naive label input (NLI) mechanism and an improved label input (ILI) mechanism to incorporate regression labels into the generator and the discriminator. The reformulation in (S1) leads to two novel empirical discriminator losses, termed the hard vicinal discriminator loss (HVDL) and the soft vicinal discriminator loss (SVDL) respectively, and a novel empirical generator loss. Hence, we propose four versions of CcGAN employing different proposed losses and label input mechanisms. The error bounds of the discriminator trained with HVDL and SVDL, respectively, are derived under mild assumptions. To evaluate the performance of CcGANs, two new benchmark datasets (RC-49 and Cell-200) are created. A novel evaluation metric (Sliding Fréchet Inception Distance) is also proposed to replace Intra-FID when Intra-FID is not applicable. Our extensive experiments on several benchmark datasets (i.e., RC-49, UTKFace, Cell-200, and Steering Angle with both low and high resolutions) support the following findings: the proposed CcGAN is able to generate diverse, high-quality samples from the image distribution conditional on a given regression label; and CcGAN substantially outperforms cGAN both visually and quantitatively.

A Bayesian coreset is a small, weighted subset of data that replaces the full dataset during Bayesian inference, with the goal of reducing computational cost. Although past work has shown empirically that there often exists a coreset with low inferential error, efficiently constructing such a coreset remains a challenge. Current methods tend to be slow, require a secondary inference step after coreset construction, and do not provide bounds on the data marginal evidence. In this work, we introduce a new method---sparse Hamiltonian flows---that addresses all three of these challenges. The method involves first subsampling the data uniformly, and then optimizing a Hamiltonian flow parametrized by coreset weights and including periodic momentum quasi-refreshment steps. Theoretical results show that the method enables an exponential compression of the dataset in a representative model, and that the quasi-refreshment steps reduce the KL divergence to the target. Real and synthetic experiments demonstrate that sparse Hamiltonian flows provide accurate posterior approximations with significantly reduced runtime compared with competing dynamical-system-based inference methods.

Standard Bayesian inference algorithms are prohibitively expensive in the regime of modern large-scale data. Recent work has found that a small, weighted subset of data (a coreset) may be used in place of the full dataset during inference, taking advantage of data redundancy to reduce computational cost. However, this approach has limitations in the increasingly common setting of sensitive, high-dimensional data. Indeed, we prove that there are situations in which the Kullback-Leibler divergence between the optimal coreset and the true posterior grows with data dimension; and as coresets include a subset of the original data, they cannot be constructed in a manner that preserves individual privacy. We address both of these issues with a single unified solution, Bayesian pseudocoresets --- a small weighted collection of synthetic "pseudodata"---along with a variational optimization method to select both pseudodata and weights. The use of pseudodata (as opposed to the original datapoints) enables both the summarization of high-dimensional data and the differentially private summarization of sensitive data. Real and synthetic experiments on high-dimensional data demonstrate that Bayesian pseudocoresets achieve significant improvements in posterior approximation error compared to traditional coresets, and that pseudocoresets provide privacy without a significant loss in approximation quality.