We introduce beta diffusion, a novel generative modeling method that integrates demasking and denoising to generate data within bounded ranges. Using scaled and shifted beta distributions, beta diffusion utilizes multiplicative transitions over time to create both forward and reverse diffusion processes, maintaining beta distributions in both the forward marginals and the reverse conditionals, given the data at any point in time. Unlike traditional diffusion-based generative models relying on additive Gaussian noise and reweighted evidence lower bounds (ELBOs), beta diffusion is multiplicative and optimized with KL-divergence upper bounds (KLUBs) derived from the convexity of the KL divergence. We demonstrate that the proposed KLUBs are more effective for optimizing beta diffusion compared to negative ELBOs, which can also be derived as the KLUBs of the same KL divergence with its two arguments swapped. The loss function of beta diffusion, expressed in terms of Bregman divergence, further supports the efficacy of KLUBs for optimization. Experimental results on both synthetic data and natural images demonstrate the unique capabilities of beta diffusion in generative modeling of range-bounded data and validate the effectiveness of KLUBs in optimizing diffusion models, thereby making them valuable additions to the family of diffusion-based generative models and the optimization techniques used to train them.
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Many problems can be viewed as forms of geospatial search aided by aerial imagery, with examples ranging from detecting poaching activity to human trafficking. We model this class of problems in a visual active search (VAS) framework, which has three key inputs: (1) an image of the entire search area, which is subdivided into regions, (2) a local search function, which determines whether a previously unseen object class is present in a given region, and (3) a fixed search budget, which limits the number of times the local search function can be evaluated. The goal is to maximize the number of objects found within the search budget. We propose a reinforcement learning approach for VAS that learns a meta-search policy from a collection of fully annotated search tasks. This meta-search policy is then used to dynamically search for a novel target-object class, leveraging the outcome of any previous queries to determine where to query next. Through extensive experiments on several large-scale satellite imagery datasets, we show that the proposed approach significantly outperforms several strong baselines. We also propose novel domain adaptation techniques that improve the policy at decision time when there is a significant domain gap with the training data. Code is publicly available.
Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear time-invariant (LTI) ODEs -- turning multi-step forecasting into sparse matrix multiplications. Though there exists a variety of learning approaches, they usually lack crucial learning-theoretic guarantees, making the behavior of the obtained models with increasing data and dimensionality unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) over trajectories that solely spans transformations into LTI dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization error bounds under weaker assumptions than existing work. Our experiments demonstrate superior forecasting performance compared to Koopman operator and sequential data predictors in RKHS.
Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling problems on rectangular domains. To lift such a restriction and permit FFT on irregular geometries as well as topology changes, we introduce domain agnostic Fourier neural operator (DAFNO), a novel neural operator architecture for learning surrogates with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of FNOs, and leverage FFT to achieve rapid computations, in such a way that the geometric information is explicitly encoded in the architecture. In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as compared to baseline neural operator models on two benchmark datasets of material modeling and airfoil simulation. To further demonstrate the capability and generalizability of DAFNO in handling complex domains with topology changes, we consider a brittle material fracture evolution problem. With only one training crack simulation sample, DAFNO has achieved generalizability to unseen loading scenarios and substantially different crack patterns from the trained scenario. Our code and data accompanying this paper are available at //github.com/ningliu-iga/DAFNO.
Agents with the ability to comprehend and reason about the dynamics of objects would be expected to exhibit improved robustness and generalization in novel scenarios. However, achieving this capability necessitates not only an effective scene representation but also an understanding of the mechanisms governing interactions among object subsets. Recent studies have made significant progress in representing scenes using object slots. In this work, we introduce Reusable Slotwise Mechanisms, or RSM, a framework that models object dynamics by leveraging communication among slots along with a modular architecture capable of dynamically selecting reusable mechanisms for predicting the future states of each object slot. Crucially, RSM leverages the Central Contextual Information (CCI), enabling selected mechanisms to access the remaining slots through a bottleneck, effectively allowing for modeling of higher order and complex interactions that might require a sparse subset of objects. Experimental results demonstrate the superior performance of RSM compared to state-of-the-art methods across various future prediction and related downstream tasks, including Visual Question Answering and action planning. Furthermore, we showcase RSM's Out-of-Distribution generalization ability to handle scenes in intricate scenarios.
We design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability from Impagliazzo et al. [2022]. According to this definition, a clustering algorithm is replicable if, with high probability, its output induces the exact same partition of the sample space after two executions on different inputs drawn from the same distribution, when its internal randomness is shared across the executions. We propose such algorithms for the statistical $k$-medians, statistical $k$-means, and statistical $k$-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable $O(1)$-approximation algorithm for statistical Euclidean $k$-medians ($k$-means) with $\operatorname{poly}(d)$ sample complexity. We also describe an $O(1)$-approximation algorithm with an additional $O(1)$-additive error for statistical Euclidean $k$-centers, albeit with $\exp(d)$ sample complexity. In addition, we provide experiments on synthetic distributions in 2D using the $k$-means++ implementation from sklearn as a black-box that validate our theoretical results.
We generalize the continuous time framework for score-based generative models from an underlying Brownian motion (BM) to an approximation of fractional Brownian motion (FBM). We derive a continuous reparameterization trick and the reverse time model by representing FBM as a stochastic integral over a family of Ornstein-Uhlenbeck processes to define generative fractional diffusion models (GFDM) with driving noise converging to a non-Markovian process of infinite quadratic variation. The Hurst index $H\in(0,1)$ of FBM enables control of the roughness of the distribution transforming path. To the best of our knowledge, this is the first attempt to build a generative model upon a stochastic process with infinite quadratic variation.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
The information bottleneck (IB) method is a technique for extracting information that is relevant for predicting the target random variable from the source random variable, which is typically implemented by optimizing the IB Lagrangian that balances the compression and prediction terms. However, the IB Lagrangian is hard to optimize, and multiple trials for tuning values of Lagrangian multiplier are required. Moreover, we show that the prediction performance strictly decreases as the compression gets stronger during optimizing the IB Lagrangian. In this paper, we implement the IB method from the perspective of supervised disentangling. Specifically, we introduce Disentangled Information Bottleneck (DisenIB) that is consistent on compressing source maximally without target prediction performance loss (maximum compression). Theoretical and experimental results demonstrate that our method is consistent on maximum compression, and performs well in terms of generalization, robustness to adversarial attack, out-of-distribution detection, and supervised disentangling.
We investigate a lattice-structured LSTM model for Chinese NER, which encodes a sequence of input characters as well as all potential words that match a lexicon. Compared with character-based methods, our model explicitly leverages word and word sequence information. Compared with word-based methods, lattice LSTM does not suffer from segmentation errors. Gated recurrent cells allow our model to choose the most relevant characters and words from a sentence for better NER results. Experiments on various datasets show that lattice LSTM outperforms both word-based and character-based LSTM baselines, achieving the best results.
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