In this work, we propose Functional Flow Matching (FFM), a function-space generative model that generalizes the recently-introduced Flow Matching model to operate directly in infinite-dimensional spaces. Our approach works by first defining a path of probability measures that interpolates between a fixed Gaussian measure and the data distribution, followed by learning a vector field on the underlying space of functions that generates this path of measures. Our method does not rely on likelihoods or simulations, making it well-suited to the function space setting. We provide both a theoretical framework for building such models and an empirical evaluation of our techniques. We demonstrate through experiments on synthetic and real-world benchmarks that our proposed FFM method outperforms several recently proposed function-space generative models.
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Flow matching is a recent framework to train generative models that exhibits impressive empirical performance while being relatively easier to train compared with diffusion-based models. Despite its advantageous properties, prior methods still face the challenges of expensive computing and a large number of function evaluations of off-the-shelf solvers in the pixel space. Furthermore, although latent-based generative methods have shown great success in recent years, this particular model type remains underexplored in this area. In this work, we propose to apply flow matching in the latent spaces of pretrained autoencoders, which offers improved computational efficiency and scalability for high-resolution image synthesis. This enables flow-matching training on constrained computational resources while maintaining their quality and flexibility. Additionally, our work stands as a pioneering contribution in the integration of various conditions into flow matching for conditional generation tasks, including label-conditioned image generation, image inpainting, and semantic-to-image generation. Through extensive experiments, our approach demonstrates its effectiveness in both quantitative and qualitative results on various datasets, such as CelebA-HQ, FFHQ, LSUN Church & Bedroom, and ImageNet. We also provide a theoretical control of the Wasserstein-2 distance between the reconstructed latent flow distribution and true data distribution, showing it is upper-bounded by the latent flow matching objective. Our code will be available at //github.com/VinAIResearch/LFM.git.
This article explains the usage of R package CausalModels, which is publicly available on the Comprehensive R Archive Network. While packages are available for sufficiently estimating causal effects, there lacks a package that provides a collection of structural models using the conventional statistical approach developed by Hern\'an and Robins (2020). CausalModels addresses this deficiency of software in R concerning causal inference by offering tools for methods that account for biases in observational data without requiring extensive statistical knowledge. These methods should not be ignored and may be more appropriate or efficient in solving particular problems. While implementations of these statistical models are distributed among a number of causal packages, CausalModels introduces a simple and accessible framework for a consistent modeling pipeline among a variety of statistical methods for estimating causal effects in a single R package. It consists of common methods including standardization, IP weighting, G-estimation, outcome regression, instrumental variables and propensity matching.
We take a random matrix theory approach to random sketching and show an asymptotic first-order equivalence of the regularized sketched pseudoinverse of a positive semidefinite matrix to a certain evaluation of the resolvent of the same matrix. We focus on real-valued regularization and extend previous results on an asymptotic equivalence of random matrices to the real setting, providing a precise characterization of the equivalence even under negative regularization, including a precise characterization of the smallest nonzero eigenvalue of the sketched matrix, which may be of independent interest. We then further characterize the second-order equivalence of the sketched pseudoinverse. We also apply our results to the analysis of the sketch-and-project method and to sketched ridge regression. Lastly, we prove that these results generalize to asymptotically free sketching matrices, obtaining the resulting equivalence for orthogonal sketching matrices and comparing our results to several common sketches used in practice.
In generative modeling, numerous successful approaches leverage a low-dimensional latent space, e.g., Stable Diffusion models the latent space induced by an encoder and generates images through a paired decoder. Although the selection of the latent space is empirically pivotal, determining the optimal choice and the process of identifying it remain unclear. In this study, we aim to shed light on this under-explored topic by rethinking the latent space from the perspective of model complexity. Our investigation starts with the classic generative adversarial networks (GANs). Inspired by the GAN training objective, we propose a novel "distance" between the latent and data distributions, whose minimization coincides with that of the generator complexity. The minimizer of this distance is characterized as the optimal data-dependent latent that most effectively capitalizes on the generator's capacity. Then, we consider parameterizing such a latent distribution by an encoder network and propose a two-stage training strategy called Decoupled Autoencoder (DAE), where the encoder is only updated in the first stage with an auxiliary decoder and then frozen in the second stage while the actual decoder is being trained. DAE can improve the latent distribution and as a result, improve the generative performance. Our theoretical analyses are corroborated by comprehensive experiments on various models such as VQGAN and Diffusion Transformer, where our modifications yield significant improvements in sample quality with decreased model complexity.
Variational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics. Typically, VI fits a simple parametric distribution to the target posterior by minimizing an appropriate objective such as the evidence lower bound (ELBO). In this work, we present a new approach to VI based on the principle of score matching, that if two distributions are equal then their score functions (i.e., gradients of the log density) are equal at every point on their support. With this, we develop score matching VI, an iterative algorithm that seeks to match the scores between the variational approximation and the exact posterior. At each iteration, score matching VI solves an inner optimization, one that minimally adjusts the current variational estimate to match the scores at a newly sampled value of the latent variables. We show that when the variational family is a Gaussian, this inner optimization enjoys a closed form solution, which we call Gaussian score matching VI (GSM-VI). GSM-VI is also a ``black box'' variational algorithm in that it only requires a differentiable joint distribution, and as such it can be applied to a wide class of models. We compare GSM-VI to black box variational inference (BBVI), which has similar requirements but instead optimizes the ELBO. We study how GSM-VI behaves as a function of the problem dimensionality, the condition number of the target covariance matrix (when the target is Gaussian), and the degree of mismatch between the approximating and exact posterior distribution. We also study GSM-VI on a collection of real-world Bayesian inference problems from the posteriorDB database of datasets and models. In all of our studies we find that GSM-VI is faster than BBVI, but without sacrificing accuracy. It requires 10-100x fewer gradient evaluations to obtain a comparable quality of approximation.
Addressing the interpretability problem of NMF on Boolean data, Boolean Matrix Factorization (BMF) uses Boolean algebra to decompose the input into low-rank Boolean factor matrices. These matrices are highly interpretable and very useful in practice, but they come at the high computational cost of solving an NP-hard combinatorial optimization problem. To reduce the computational burden, we propose to relax BMF continuously using a novel elastic-binary regularizer, from which we derive a proximal gradient algorithm. Through an extensive set of experiments, we demonstrate that our method works well in practice: On synthetic data, we show that it converges quickly, recovers the ground truth precisely, and estimates the simulated rank exactly. On real-world data, we improve upon the state of the art in recall, loss, and runtime, and a case study from the medical domain confirms that our results are easily interpretable and semantically meaningful.
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular matrices, where we propose a novel computational approach based on combinatorial techniques for finding the inverse of a general non-singular triangular matrix. Unlike iterative methods, our combinatorial approach for (block) triangular-type matrices enables direct computation of the matrix inverse through a nonlinear combination of carefully selected combinatorial entries from the initial matrix. This unique characteristic makes our proposed method fully parallelizable, offering significant potential for efficient implementation on parallel computing architectures. Our approach demonstrates intriguing features that allow the derivation of recurrent relations for constructing the matrix inverse. By combining the (block) combinatorial approach, with a recursive triangular split method for inverting triangular matrices, we develop potentially competitive algorithms that strike a balance between efficiency and accuracy. We provide rigorous mathematical proofs of the newly presented method. Additionally, we conduct extensive numerical tests to showcase its applicability and efficiency. The comprehensive evaluation and experimental results presented in this paper confirm the practical utility of our proposed algorithms, demonstrating their superiority over classical approaches in terms of computational efficiency.
A regular expression specifies a set of strings formed by single characters combined with concatenation, union, and Kleene star operators. Given a regular expression $R$ and a string $Q$, the regular expression matching problem is to decide if $Q$ matches any of the strings specified by $R$. Regular expressions are a fundamental concept in formal languages and regular expression matching is a basic primitive for searching and processing data. A standard textbook solution [Thompson, CACM 1968] constructs and simulates a nondeterministic finite automaton, leading to an $O(nm)$ time algorithm, where $n$ is the length of $Q$ and $m$ is the length of $R$. Despite considerable research efforts only polylogarithmic improvements of this bound are known. Recently, conditional lower bounds provided evidence for this lack of progress when Backurs and Indyk [FOCS 2016] proved that, assuming the strong exponential time hypothesis (SETH), regular expression matching cannot be solved in $O((nm)^{1-\epsilon})$, for any constant $\epsilon > 0$. Hence, the complexity of regular expression matching is essentially settled in terms of $n$ and $m$. In this paper, we take a new approach and go beyond worst-case analysis in $n$ and $m$. We introduce a \emph{density} parameter, $\Delta$, that captures the amount of nondeterminism in the NFA simulation on $Q$. The density is at most $nm+1$ but can be significantly smaller. Our main result is a new algorithm that solves regular expression matching in $$O\left(\Delta \log \log \frac{nm}{\Delta} +n + m\right)$$ time. This essentially replaces $nm$ with $\Delta$ in the complexity of regular expression matching. We complement our upper bound by a matching conditional lower bound that proves that we cannot solve regular expression matching in time $O(\Delta^{1-\epsilon})$ for any constant $\epsilon > 0$ assuming SETH.
In the domain generalization literature, a common objective is to learn representations independent of the domain after conditioning on the class label. We show that this objective is not sufficient: there exist counter-examples where a model fails to generalize to unseen domains even after satisfying class-conditional domain invariance. We formalize this observation through a structural causal model and show the importance of modeling within-class variations for generalization. Specifically, classes contain objects that characterize specific causal features, and domains can be interpreted as interventions on these objects that change non-causal features. We highlight an alternative condition: inputs across domains should have the same representation if they are derived from the same object. Based on this objective, we propose matching-based algorithms when base objects are observed (e.g., through data augmentation) and approximate the objective when objects are not observed (MatchDG). Our simple matching-based algorithms are competitive to prior work on out-of-domain accuracy for rotated MNIST, Fashion-MNIST, PACS, and Chest-Xray datasets. Our method MatchDG also recovers ground-truth object matches: on MNIST and Fashion-MNIST, top-10 matches from MatchDG have over 50% overlap with ground-truth matches.
To retrieve more relevant, appropriate and useful documents given a query, finding clues about that query through the text is crucial. Recent deep learning models regard the task as a term-level matching problem, which seeks exact or similar query patterns in the document. However, we argue that they are inherently based on local interactions and do not generalise to ubiquitous, non-consecutive contextual relationships.In this work, we propose a novel relevance matching model based on graph neural networks to leverage the document-level word relationships for ad-hoc retrieval. In addition to the local interactions, we explicitly incorporate all contexts of a term through the graph-of-word text format. Matching patterns can be revealed accordingly to provide a more accurate relevance score. Our approach significantly outperforms strong baselines on two ad-hoc benchmarks. We also experimentally compare our model with BERT and show our ad-vantages on long documents.
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