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Applications · Neuroscience, Cognitive Science

Amin Nejatbakhsh, Victor Geadah, Alex Williams, David Lipshutz

Biological and artificial neural systems form high-dimensional neural representations that underpin their computational capabilities. Methods for quantifying geometric similarity in neural representations have become a popular tool for identifying computational principles that are potentially shared across neural systems. These methods generally assume that neural responses are deterministic and static. However, responses of biological systems, and some artificial systems, are noisy and dynamically unfold over time. Furthermore, these characteristics can have substantial influence on a system’s computational capabilities. Here, we demonstrate that existing metrics can fail to capture key differences between neural systems with noisy dynamic responses. We then propose a metric for comparing the geometry of noisy neural trajectories, which can be derived as an optimal transport distance between Gaussian processes. We use the metric to compare models of neural responses in different regions of the motor system and to compare the dynamics of latent diffusion models for text-to-image synthesis.

Theory · Probabilistic Methods

Peiyao Lai, Oren Mangoubi

Given a matrix $A \in \mathbb{R}^{m\times d}$ with singular values $\sigma_1\geq \cdots \geq \sigma_d$, and a random matrix $G \in \mathbb{R}^{m\times d}$ with iid $N(0,T)$ entries for some $T>0$, we derive new bounds on the Frobenius distance between subspaces spanned by the top-$k$ (right) singular vectors of $A$ and $A+G$. This problem arises in numerous applications in statistics where a data matrix may be corrupted by Gaussian noise, and in the analysis of the Gaussian mechanism in differential privacy, where Gaussian noise is added to data to preserve private information. We show that, for matrices $A$ where the gaps in the top-$k$ singular values are roughly $\Omega(\sigma_k-\sigma_{k+1})$ the expected Frobenius distance between the subspaces is $\tilde{O}(\frac{\sqrt{d}}{\sigma_k-\sigma_{k+1}} \times \sqrt{T})$, improving on previous bounds by a factor of $\frac{\sqrt{m}}{\sqrt{d}}$. To obtain our bounds we view the perturbation to the singular vectors as a diffusion process-- the Dyson-Bessel process-- and use tools from stochastic calculus to track the evolution of the subspace spanned by the top-$k$ singular vectors, which may be of independent interest.

Optimization · Zero-order and Black-box Optimization

Yanbo Zhang, Benedikt Hartl, Hananel Hazan, Michael Levin

In a convergence of machine learning and biology, we reveal that diffusion models are evolutionary algorithms. By considering evolution as a denoising process and reversed evolution as diffusion, we mathematically demonstrate that diffusion models inherently perform evolutionary algorithms, naturally encompassing selection, mutation, and reproductive isolation. Building on this equivalence, we propose the Diffusion Evolution method: an evolutionary algorithm utilizing iterative denoising -- as originally introduced in the context of diffusion models -- to heuristically refine solutions in parameter spaces. Unlike traditional approaches, Diffusion Evolution efficiently identifies multiple optimal solutions and outperforms prominent mainstream evolutionary algorithms. Furthermore, leveraging advanced concepts from diffusion models, namely latent space diffusion and accelerated sampling, we introduce Latent Space Diffusion Evolution, which finds solutions for evolutionary tasks in high-dimensional complex parameter space while significantly reducing computational steps. This parallel between diffusion and evolution not only bridges two different fields but also opens new avenues for mutual enhancement, raising questions about open-ended evolution and potentially utilizing non-Gaussian or discrete diffusion models in the context of Diffusion Evolution.

Theory · Probabilistic Methods

Yibo Yang, Justus Will, Stephan Mandt

Diffusion probabilistic models have achieved mainstream success in many generative modeling tasks, from image generation to inverse problem solving. A distinct feature of these models is that they correspond to deep hierarchical latent variable models optimizing a variational evidence lower bound (ELBO) on the data likelihood. Drawing on a basic connection between likelihood modeling and compression, we explore the potential of diffusion models for progressive coding, resulting in a sequence of bits that can be incrementally transmitted and decoded with progressively improving reconstruction quality. Unlike prior work based on Gaussian diffusion or conditional diffusion models, we propose a new form of diffusion model with uniform noise in the forward process, whose negative ELBO corresponds to the end-to-end compression cost using universal quantization. We obtain promising first results on image compression, achieving competitive rate-distortion-realism results on a wide range of bit-rates with a single model, bringing neural codecs a step closer to practical deployment. Our code can be found at https://github.com/mandt-lab/uqdm.

Theory · Probabilistic Methods

Yibo Yang, Justus Will, Stephan Mandt

Diffusion probabilistic models have achieved mainstream success in many generative modeling tasks, from image generation to inverse problem solving. A distinct feature of these models is that they correspond to deep hierarchical latent variable models optimizing a variational evidence lower bound (ELBO) on the data likelihood. Drawing on a basic connection between likelihood modeling and compression, we explore the potential of diffusion models for progressive coding, resulting in a sequence of bits that can be incrementally transmitted and decoded with progressively improving reconstruction quality. Unlike prior work based on Gaussian diffusion or conditional diffusion models, we propose a new form of diffusion model with uniform noise in the forward process, whose negative ELBO corresponds to the end-to-end compression cost using universal quantization. We obtain promising first results on image compression, achieving competitive rate-distortion-realism results on a wide range of bit-rates with a single model, bringing neural codecs a step closer to practical deployment. Our code can be found at https://github.com/mandt-lab/uqdm.

Deep Learning · Generative Models and Autoencoders

Dario Shariatian, Umut Simsekli, Alain Oliviero Durmus

Investigating noise distributions beyond Gaussian in diffusion generative models remains an open challenge. The Gaussian case has been a large success experimentally and theoretically, admitting a unified stochastic differential equation (SDE) framework, encompassing score-based and denoising formulations. Recent studies have investigated the potential of \emph{heavy-tailed} noise distributions to mitigate mode collapse and effectively manage datasets exhibiting class imbalance, heavy tails, or prominent outliers. Very recently, Yoon et al.\ (NeurIPS 2023), presented the Levy-Ito model (LIM), directly extending the SDE-based framework to a class of heavy-tailed SDEs, where the injected noise followed an $\alpha$-stable distribution -- a rich class of heavy-tailed distributions. Despite its theoretical elegance and performance improvements, LIM relies on highly involved mathematical techniques, which may limit its accessibility and hinder its broader adoption and further development. In this study, we take a step back, and instead of starting from the SDE formulation, we extend the denoising diffusion probabilistic model (DDPM) by directly replacing the Gaussian noise with $\alpha$-stable noise. By using only elementary proof techniques, we show that the proposed approach, \emph{denoising L\'{e}vy probabilistic model} (DLPM) algorithmically boils down to running vanilla DDPM with minor modifications, hence allowing the use of existing implementations with minimal changes. Remarkably, as opposed to the Gaussian case, DLPM and LIM yield different training algorithms and different backward processes, leading to distinct sampling algorithms. This fundamental difference translates favorably for the performance of DLPM in various aspects: our experiments show that DLPM achieves better coverage of the tails of the data distribution, better generation of unbalanced datasets, and improved computation times requiring significantly smaller number of backward steps.

Applications · Computer Vision

Isabella Liu, Hao Su, Xiaolong Wang

Modern 3D engines and graphics pipelines require mesh as a memory-efficient representation, which allows efficient rendering, geometry processing, texture editing, and many other downstream operations. However, it is still highly difficult to obtain high-quality mesh in terms of detailed structure and time consistency from dynamic observations. To this end, we introduce Dynamic Gaussians Mesh (DG-Mesh), a framework to reconstruct a high-fidelity and time-consistent mesh from dynamic input. Our work leverages the recent advancement in 3D Gaussian Splatting to construct the mesh sequence with temporal consistency from dynamic observations. Building on top of this representation, DG-Mesh recovers high-quality meshes from the Gaussian points and can track the mesh vertices over time, which enables applications such as texture editing on dynamic objects. We introduce the Gaussian-Mesh Anchoring, which encourages evenly distributed Gaussians, resulting better mesh reconstruction through mesh-guided densification and pruning on the deformed Gaussians. By applying cycle-consistent deformation between the canonical and the deformed space, we can project the anchored Gaussian back to the canonical space and optimize Gaussians across all time frames. During the evaluation on different datasets, DG-Mesh provides significantly better mesh reconstruction and rendering than baselines.

Deep Learning · Generative Models and Autoencoders

Enshu Liu, Xuefei Ning, Yu Wang, Zinan Lin

Autoregressive (AR) models have recently achieved state-of-the-art performance in text and image generation. However, their primary limitation is slow generation speed due to the token-by-token process. We ask an ambitious question: can a pre-trained AR model be adapted to generate outputs in just one or two steps? If successful, this would significantly advance the development and deployment of AR models. We notice that existing works that attempt to speed up AR generation by generating multiple tokens at once fundamentally cannot capture the output distribution due to the conditional dependencies between tokens, limiting their effectiveness for few-step generation. To overcome this, we propose Distilled Decoding (DD), which leverages flow matching to create a deterministic mapping from Gaussian distribution to the output distribution of the pre-trained AR model. We then train a network to distill this mapping, enabling few-step generation. The entire training process of DD does not need the training data of the original AR model (as opposed to some other methods), thus making DD more practical. We evaluate DD on state-of-the-art image AR models and present promising results. For VAR, which requires 10-step generation (680 tokens), DD enables one-step generation (6.3$\times$ speed-up), with an acceptable increase in FID from 4.19 to 9.96. Similarly, for LlamaGen, DD reduces generation from 256 steps to 1, achieving an 217.8$\times$ speed-up with a comparable FID increase from 4.11 to 11.35. In both cases, baseline methods completely fail with FID scores $>$100. As the first work to demonstrate the possibility of one-step generation for image AR models, DD challenges the prevailing notion that AR models are inherently slow, and opens up new opportunities for efficient AR generation. The code and the pre-trained models will be released at https://github.com/imagination-research/distilled-decoding. The project website is at https://imagination-research.github.io/distilled-decoding.

Probabilistic Methods · Gaussian Processes

Jorge Loría, Anindya Bhadra

Neal (1996) proved that infinitely wide shallow Bayesian neural networks (BNN) converge to Gaussian processes (GP), when the network weights have bounded prior variance. Cho & Saul (2009) provided a useful recursive formula for deep kernel processes for relating the covariance kernel of each layer to the layer immediately below. Moreover, they worked out the form of the layer-wise covariance kernel in an explicit manner for several common activation functions, including the ReLU. Subsequent works have made the connection between these two works, and provided useful results on the covariance kernel of a deep GP arising as wide limits of various deep Bayesian network architectures. However, recent works, including Aitchison et al. (2021), have highlighted that the covariance kernels obtained in this manner are deterministic and hence, precludes any possibility of representation learning, which amounts to learning a non-degenerate posterior of a random kernel given the data. To address this, they propose adding artificial noise to the kernel to retain stochasticity, and develop deep kernel Wishart and inverse Wishart processes. Nonetheless, this artificial noise injection could be critiqued in that it would not naturally emerge in a classic BNN architecture under an infinite-width limit. To address this, we show that a Bayesian deep neural network, where each layer width approaches infinity, and all network weights are elliptically distributed with infinite variance, converges to a process with $\alpha$-stable marginals in each layer that has a conditionally Gaussian representation. These conditional random covariance kernels could be recursively linked in the manner of Cho & Saul (2009), even though marginally the process exhibits stable behavior, and hence covariances are not even necessarily defined. We also provide useful generalizations of the recent results of Loría & Bhadra (2024) on shallow networks to multi-layer networks, and remedy the prohibitive computational burden of their approach. The computational and statistical benefits over competing approaches stand out in simulations and in demonstrations on benchmark data sets.

Deep Learning · Generative Models and Autoencoders

Beomsu Kim, Jaemin Kim, Jeongsol Kim, Jong Chul YE

Diffusion-based generative models excel in unconditional generation, as well as on applied tasks such as image editing and restoration. The success of diffusion models lies in the iterative nature of diffusion: diffusion breaks down the complex process of mapping noise to data into a sequence of simple denoising tasks. Moreover, we are able to exert fine-grained control over the generation process by injecting guidance terms into each denoising step. However, the iterative process is also computationally intensive, often taking from tens up to thousands of function evaluations. Although consistency trajectory models (CTMs) enable traversal between any time points along the probability flow ODE (PFODE) and score inference with a single function evaluation, CTMs only allow translation from Gaussian noise to data. Thus, this work aims to unlock the full potential of CTMs by proposing generalized CTMs (GCTMs), which translate between arbitrary distributions via ODEs. We discuss the design space of GCTMs and demonstrate their efficacy in various image manipulation tasks such as image-to-image translation, restoration, and editing. Code is available at https://github.com/1202kbs/GCTM.

Theory · Probabilistic Methods

Xinyu Zhang, Daolang Huang, Samuel Kaski, Julien Martinelli

Preferential Bayesian Optimization (PBO) is a sample-efficient method to learn latent user utilities from preferential feedback over a pair of designs. It relies on a statistical surrogate model for the latent function, usually a Gaussian process, and an acquisition strategy to select the next candidate pair to get user feedback on. Due to the non-conjugacy of the associated likelihood, every PBO step requires a significant amount of computations with various approximate inference techniques. This computational overhead is incompatible with the way humans interact with computers, hindering the use of PBO in real-world cases. Building on the recent advances of amortized BO, we propose to circumvent this issue by fully amortizing PBO, meta-learning both the surrogate and the acquisition function. Our method comprises a novel transformer neural process architecture, trained using reinforcement learning and tailored auxiliary losses.On a benchmark composed of synthetic and real-world datasets, our method is several orders of magnitude faster than the usual Gaussian process-based strategies and often outperforms them in accuracy.

Xiaoqiang Lin, Xinyi Xu, See-Kiong Ng, Bryan Kian Hsiang Low

Data valuation has found many real-world applications, e.g., data pricing and data selection. However, the most adopted approach -- Shapley value (SV) -- is computationally expensive due to the large number of model trainings required. Fortunately, most applications (e.g., data selection) require only knowing the $m$ data points with the highest data values (i.e., top-$m$ data values), which implies the potential for fewer model trainings as exact data values are not required. Existing work formulates top-$m$ Shapley value identification as top-$m$ arms identification in multi-armed bandits (MAB). However, the proposed approach falls short because it does not utilize data features to predict data values, a method that has been shown empirically to be effective. A recent top-$m$ arms identification work does consider the use of arm features while assuming a linear relationship between arm features and rewards, which is often not satisfied in data valuation. To this end, we propose the GPGapE algorithm that uses the Gaussian process to model the \emph{non-linear} mapping from data features to data values, removing the linear assumption. We theoretically analyze the correctness and stopping iteration of GPGapE in finding an $(\epsilon, \delta)$-approximation to the top-$m$ data values. We further improve the computational efficiency, by calculating data values using small data subsets to reduce the computation cost of model training. We empirically demonstrate that GPGapE outperforms other baselines in top-$m$ data values identification, noisy data detection, and data subset selection on real-world datasets. We also demonstrate the efficiency of our GPGapE in data selection for large language model fine-tuning.

Theory · Online Learning and Bandits

Jack Sandberg, Niklas Åkerblom, Morteza Haghir Chehreghani

We consider the combinatorial volatile Gaussian process (GP) semi-bandit problem. Each round, an agent is provided a set of available base arms and must select a subset of them to maximize the long-term cumulative reward. We study the Bayesian setting and provide novel Bayesian cumulative regret bounds for three GP-based algorithms: GP-UCB, GP-BayesUCB and GP-TS. Our bounds extend previous results for GP-UCB and GP-TS to the \emph{infinite}, \emph{volatile} and \emph{combinatorial} setting, and to the best of our knowledge, we provide the first regret bound for GP-BayesUCB. Volatile arms encompass other widely considered bandit problems such as contextual bandits.Furthermore, we employ our framework to address the challenging real-world problem of online energy-efficient navigation, where we demonstrate its effectiveness compared to the alternatives.

Optimization · Optimization and Learning under Uncertainty

Paul Brunzema, Mikkel Jordahn, John Willes, Sebastian Trimpe, Jasper Snoek, James Harrison

Gaussian Processes (GPs) are widely seen as the state-of-the-art surrogate models for Bayesian optimization (BO) due to their ability to model uncertainty and their performance on tasks where correlations are easily captured (such as those defined by Euclidean metrics) and their ability to be efficiently updated online. However, the performance of GPs depends on the choice of kernel, and kernel selection for complex correlation structures is often difficult or must be made bespoke. While Bayesian neural networks (BNNs) are a promising direction for higher capacity surrogate models, they have so far seen limited use due to poor performance on some problem types. In this paper, we propose an approach which shows competitive performance on many problem types, including some that BNNs typically struggle with. We build on variational Bayesian last layers (VBLLs), and connect training of these models to exact conditioning in GPs. We exploit this connection to develop an efficient online training algorithm that interleaves conditioning and optimization. Our findings suggest that VBLL networks significantly outperform GPs and other BNN architectures on tasks with complex input correlations, and match the performance of well-tuned GPs on established benchmark tasks.

Probabilistic Methods · Monte Carlo and Sampling Methods

Isaac Reid, Stratis Markou, Krzysztof Choromanski, Richard E Turner, Adrian Weller

Random features (RFs) are a popular technique to scale up kernel methods in machine learning, replacing exact kernel evaluations with stochastic Monte Carlo estimates. They underpin models as diverse as efficient transformers (by approximating attention) to sparse spectrum Gaussian processes (by approximating the covariance function). Efficiency can be further improved by speeding up the convergence of these estimates: a variance reduction problem. We tackle this through the unifying lens of optimal transport, finding couplings to improve RFs defined on both Euclidean and discrete input spaces. They enjoy theoretical guarantees and sometimes provide strong downstream gains, including for scalable inference on graphs. We reach surprising conclusions about the benefits and limitations of variance reduction as a paradigm, showing that other properties of the coupling should be optimised for attention estimation in efficient transformers.

Deep Learning · Generative Models and Autoencoders

Vadim Popov, Assel Yermekova, Tasnima Sadekova, Artem Khrapov, Mikhail Kudinov

Lévy-Itô denoising diffusion models relying on isotropic α-stable noise instead of Gaussian distribution have recently been shown to improve performance of conventional diffusion models in image generation on imbalanced datasets while performing comparably in the standard settings. However, the stochastic algorithm of sampling from such models consists in solving the stochastic differential equation describing only an approximate inverse of the process of adding α-stable noise to data which may lead to suboptimal performance. In this paper, we derive a parametric family of stochastic differential equations whose solutions have the same marginal densities as those of the forward diffusion and show that the appropriate choice of the parameter values can improve quality of the generated images when the number of reverse diffusion steps is small. Also, we demonstrate that Lévy-Itô diffusion models are applicable to diverse domains and show that a well-trained text-to-speech Lévy-Itô model may have advantages over standard diffusion models on highly imbalanced datasets.

Theory · Optimization

Eric Anschuetz

Classical neural networks with random initialization famously behave as Gaussian processes in the limit of many neurons, which allows one to completely characterize their training and generalization behavior. No such general understanding exists for quantum neural networks (QNNs), which—outside of certain special cases—are known to not behave as Gaussian processes when randomly initialized. We here prove that QNNs and their first two derivatives instead generally form what we call "Wishart processes," where certain algebraic properties of the network determine the hyperparameters of the process. This Wishart process description allows us to, for the first time: give necessary and sufficient conditions for a QNN architecture to have a Gaussian process limit; calculate the full gradient distribution, generalizing previously known barren plateau results; and calculate the local minima distribution of algebraically constrained QNNs. Our unified framework suggests a certain simple operational definition for the "trainability" of a given QNN model using a newly introduced, experimentally accessible quantity we call the "degrees of freedom" of the network architecture.

Applications · Physics

Ann-Kathrin Schuetz, Alan Poon, Aobo Li

The experimental discovery of neutrinoless double-beta decay (NLDBD) would answer one of the most important questions in physics: Why is there more matter than antimatter in our universe? To maximize the chances of discovery, NLDBD experiments must optimize their detector designs to minimize the probability of background events contaminating the detector. Given that this probability is inherently low, design optimization either requires extremely costly simulations to generate sufficient background counts or contending with significant variance. In this work, we formalize this dilemma as a Rare Event Design (RED) problem: identifying optimal design parameters when the design metric to be minimized is inherently small. We then designed the Rare Event Surrogate Model (RESuM) for physics detector design optimization under RED conditions. RESuM uses a pre-trained Conditional Neural Process (CNP) model to incorporate additional prior knowledge into a Multi-Fidelity Gaussian Process model. We applied RESuM to optimize neutron shielding designs for the LEGEND NLDBD experiment, identifying an optimal design that reduces the neutron background by $(66.5 \pm 3.5)$% while using only 3.3% of the computational resources compared to traditional methods. Given the prevalence of RED problems in other fields of physical sciences, especially in rare-event searches, the RESuM algorithm has broad potential for accelerating simulation-intensive applications.

Applications · Computer Vision

Jiwook Kim, Seonho Lee, Jaeyo Shin, Jiho Choi, Hyunjung Shim

Score distillation sampling (SDS) has emerged as an effective framework in text-driven 3D editing tasks, leveraging diffusion models for 3D-consistent editing. However, existing SDS-based 3D editing methods suffer from long training times and produce low-quality results. We identify that the root cause of this performance degradation is their conflict with the sampling dynamics of diffusion models. Addressing this conflict allows us to treat SDS as a diffusion reverse process for 3D editing via sampling from data space. In contrast, existing methods naively distill the score function using diffusion models. From these insights, we propose DreamCatalyst, a novel framework that considers these sampling dynamics in the SDS framework. Specifically, we devise the optimization process of our DreamCatalyst to approximate the diffusion reverse process in editing tasks, thereby aligning with diffusion sampling dynamics. As a result, DreamCatalyst successfully reduces training time and improves editing quality. Our method offers two modes: (1) a fast mode that edits Neural Radiance Fields (NeRF) scenes approximately 23 times faster than current state-of-the-art NeRF editing methods, and (2) a high-quality mode that produces superior results about 8 times faster than these methods. Notably, our high-quality mode outperforms current state-of-the-art NeRF editing methods in terms of both speed and quality. DreamCatalyst also surpasses the state-of-the-art 3D Gaussian Splatting (3DGS) editing methods, establishing itself as an effective and model-agnostic 3D editing solution.

Deep Learning · Robustness

Yiming Liu, Kezhao Liu, Yao Xiao, ZiYi Dong, Xiaogang Xu, Pengxu Wei, Liang Lin

Diffusion-Based Purification (DBP) has emerged as an effective defense mechanism against adversarial attacks. The success of DBP is often attributed to the forward diffusion process, which reduces the distribution gap between clean and adversarial images by adding Gaussian noise. Although this explanation is theoretically grounded, the precise contribution of this process to robustness remains unclear. In this paper, through a systematic investigation, we propose that the intrinsic stochasticity in the DBP procedure is the primary factor driving robustness. To explore this hypothesis, we introduce a novel Deterministic White-Box (DW-box) evaluation protocol to assess robustness in the absence of stochasticity, and analyze attack trajectories and loss landscapes. Our results suggest that DBP models primarily leverage stochasticity to evade effective attack directions, and that their ability to purify adversarial perturbations can be weak. To further enhance the robustness of DBP models, we propose Adversarial Denoising Diffusion Training (ADDT), which incorporates classifier-guided adversarial perturbations into diffusion training, thereby strengthening the models' ability to purify adversarial perturbations. Additionally, we propose Rank-Based Gaussian Mapping (RBGM) to improve the compatibility of perturbations with diffusion models. Experimental results validate the effectiveness of ADDT. In conclusion, our study suggests that future research on DBP can benefit from the perspective of decoupling stochasticity-based and purification-based robustness.