Search

Latent Diffusion Models (LDMs) produce high-quality, photo-realistic images, however, the latency incurred by multiple costly inference iterations can restrict their applicability. We introduce LatentCRF, a continuous Conditional Random Field (CRF) model, implemented as a neural network layer, that models the spatial and semantic relationships among the latent vectors in the LDM. By replacing some of the computationally-intensive LDM inference iterations with our lightweight LatentCRF, we achieve a superior balance between quality, speed and diversity. We increase inference efficiency by 33% with no loss in image quality or diversity compared to the full LDM. LatentCRF is an easy add-on, which does not require modifying the LDM.

Given a dynamical system $(X, \Gamma)$, the corresponding crossed product $C^*$-algebra $C(X)\rtimes_{r}\Gamma$ is called reflecting, when every intermediate $C^*$-algebra $C^*_r(\Gamma)<\mathcal{A} < C(X)\rtimes_{r}\Gamma$ is of the form $\mathcal{A}=C(Y)\rtimes_{r}\Gamma$, corresponding to a dynamical factor $X \rightarrow Y$. It is called almost reflecting if $\mathbb{E}(\mathcal{A}) \subset \mathcal{A}$ for every such $\mathcal{A}$. These two notions coincide for groups admitting the approximation property (AP). Let $\Gamma$ be a non-elementary convergence group or a lattice in $\text{SL}_d(\mathbb{R})$ for some $d \ge 2$. We show that any uniformly rigid system $(X,\Gamma)$ is almost reflecting. In particular, this holds for any equicontinuous action. In the von Neumann setting, for the same groups $\Gamma$ and any uniformly rigid system $(X,\mathcal{B},\mu, \Gamma)$ the crossed product algebra $L^{\infty}(X,\mu)\rtimes\Gamma$ is reflecting. An inclusion of algebras $\mathcal{A}\subset\mathcal{B}$ is called $\textit{minimal ambient}$ if there are no intermediate algebras. As a demonstration of our methods, we construct examples of minimal ambient inclusions with various interesting properties in the $C^*$ and the von Neumann settings., Comment: Minor typos corrected

This paper proposes a sparse regression strategy for discovery of ordinary and partial differential equations from incomplete and noisy data. Inference is performed over both equation parameters and state variables using a statistically motivated likelihood function. Sparsity is enforced by a selection algorithm which iteratively removes terms and compares models using statistical information criteria. Large scale optimization is performed using a second-order variant of the Levenberg-Marquardt method, where the gradient and Hessian are computed via automatic differentiation. Illustrations involving canonical systems of ordinary and partial differential equations are used to demonstrate the flexibility and robustness of the approach. Accurate reconstruction of systems is found to be possible even in extreme cases of limited data and large observation noise., Comment: 19 pages, 7 figures

In this note we give a fairly direct proof of a recent theorem of Gorska, Lemanczyk and de la Rue which characterises the class of measure preserving transformations that are disjoint from every ergodic measure preserving transformation. Our proof works just as well for any countable acting group.

Let $G$ be a locally compact group and $\mu$ an admissible probability measure on $G$. Let $(B,\nu)$ be the universal topological Poisson $\mu$-boundary of $(G,\mu)$ and $\Pi_s(G)$ the universal minimal strongly proximal $G$-flow. This note is inspired by a recent result of Hartman and Kalantar. We show that for a locally compact second countable group $G$ the following conditions are equivalent: (i) the measure $\nu$ is the unique $\mu$-stationary measure on $B$, (ii) $(B,G) \cong (\Pi_s(G),G)$., Comment: The formulation of Theorem 2.1 is corrected, and a proof of corollary 2.2 is added

We consider an intermediate factor situation in two categories: probability measure preserving ergodic theory and compact topological dynamics. In the first we prove a master-key theorem and examine a wide range of applications. In the second we treat the case when one of the systems is distal and then provide some counterexamples.

As with many machine learning problems, the progress of image generation methods hinges on good evaluation metrics. One of the most popular is the Frechet Inception Distance (FID). FID estimates the distance between a distribution of Inception-v3 features of real images, and those of images generated by the algorithm. We highlight important drawbacks of FID: Inception's poor representation of the rich and varied content generated by modern text-to-image models, incorrect normality assumptions, and poor sample complexity. We call for a reevaluation of FID's use as the primary quality metric for generated images. We empirically demonstrate that FID contradicts human raters, it does not reflect gradual improvement of iterative text-to-image models, it does not capture distortion levels, and that it produces inconsistent results when varying the sample size. We also propose an alternative new metric, CMMD, based on richer CLIP embeddings and the maximum mean discrepancy distance with the Gaussian RBF kernel. It is an unbiased estimator that does not make any assumptions on the probability distribution of the embeddings and is sample efficient. Through extensive experiments and analysis, we demonstrate that FID-based evaluations of text-to-image models may be unreliable, and that CMMD offers a more robust and reliable assessment of image quality., Comment: Code is available at: https://github.com/google-research/google-research/tree/master/cmmd

Let $\Gamma$ be a countable group and $(X, \Gamma)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(\Gamma)<\mathcal{A}

We develop the theory of Patterson-Sullivan measures on the boundary of a locally compact hyperbolic group, associating to certain left invariant metrics on the group measures on the boundary. We later prove that for second countable, non-elementary, unimodular locally compact hyperbolic groups the associated Koopman representations are irreducible and their isomorphism type classifies the metric on the group up to homothety and bounded additive changes, generalizing a theorem of Garncarek on discrete hyperbolic groups. We use this to answer a question of Caprace, Kalantar and Monod on type I hyperbolic groups in the unimodular case., Comment: 43 pages

Modern text-to-image generation models produce high-quality images that are both photorealistic and faithful to the text prompts. However, this quality comes at significant computational cost: nearly all of these models are iterative and require running sampling multiple times with large models. This iterative process is needed to ensure that different regions of the image are not only aligned with the text prompt, but also compatible with each other. In this work, we propose a light-weight approach to achieving this compatibility between different regions of an image, using a Markov Random Field (MRF) model. We demonstrate the effectiveness of this method on top of the latent token-based Muse text-to-image model. The MRF richly encodes the compatibility among image tokens at different spatial locations to improve quality and significantly reduce the required number of Muse sampling steps. Inference with the MRF is significantly cheaper, and its parameters can be quickly learned through back-propagation by modeling MRF inference as a differentiable neural-network layer. Our full model, MarkovGen, uses this proposed MRF model to both speed up Muse by 1.5X and produce higher quality images by decreasing undesirable image artifacts.

In this work, on the one hand, we survey and amplify old results concerning tame dynamical systems and, on the other, prove some new results and exhibit new examples of such systems. In particular, we study tame symbolic systems and establish a neat characterization of tame subshifts. We also provide sufficient conditions which ensure that certain coding functions are tame. Finally we discuss examples where certain universal dynamical systems associated with some Polish groups are tame., Comment: 32 pages, revised proof of Theorem 6.8. arXiv admin note: substantial text overlap with arXiv:1405.2588

Our main result is to show that every infinite, countable, residually finite group $G$ admits a Hausdorff group topology which is neither discrete nor precompact.

Let $\Gamma$ be a countable group and $\mathrm{Sub}(\Gamma)$ its Chabauty space, namely the compact $\Gamma$-space consisting of all subgroups of $\Gamma$. We call a subgroup $\Delta \in \mathrm{Sub}(\Gamma)$ a boomerang subgroup if for every $\gamma \in \Gamma$, $\gamma^{n_i} \Delta \gamma^{-n_i} \rightarrow \Delta$ for some subsequence $\{n_i \} \subset \mathbb{N}$. Poincar\'{e} recurrence implies that $\mu$-almost every subgroup of $\Gamma$ is a boomerang, with respect to every invariant random subgroup $\mu$ of $\Gamma$. We establish for boomerang subgroups many density related properties, most of which are known to hold almost surely for invariant random subgroups. Let $\mathbb{K}$ be a number field, $O$ its ring of integers, $S$ a finite set of valuations including all the Archimedean valuations, and $\mathbb{G}$ an absolutely almost simple group defined over $\mathbb{K}$. Our main result is that if $\mathrm{rk}_{\mathbb{K}} \mathbb{G} \ge 2$ then any $\Gamma$ which is commensurable to the $S$-arithmetic group $\mathbb{G}(O_S)$ has very few boomerang subgroups. Namely, every boomerang in $\Gamma$ is either finite and central or of finite index. In particular we recover Margulis' normal subgroup theorem as well as the Nevo-Stuck-Zimmer theorem for such lattices. We include a short, accessible proof for the above theorem in the case that $\Gamma$ is commensurable to $\mathrm{SL}_n(\mathbb{Z}), \ n \ge 3$., Comment: 34 pages, 2 figures

Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group $G$ admits a weakly mixing probability measure preserving action $G \curvearrowright (X,\mathcal{B},\mu)$ which is faithful but not essentially free. In other words, $G$ admits a weakly mixing nontrivial faithful IRS. We also prove that every non-elementary hyperbolic group admits a characteristic random subgroup with the same properties., Comment: 14 pages, 1 figure

In this paper we study the action of a countable group $\Gamma$ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner-Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on $\mathrm{SL}_3(\mathbf{Z})$ corresponding to the semigroup of positive matrices cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended., Comment: 20 pages, 3 figures

An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In this work we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf{X}$ has zero entropy then it is not dominant.

Studying a question of Kazhdan and Yom Din we first prove some new fixed point theorems for linear actions of a discrete countable group $G$ on a dual Banach space $V^*$, and then show that the answer to the question in full generality is in the negative., Comment: An appendix by Nicolas Monod is added

We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\mathbb{Z})$ is a proper subalgebra of $\mathcal{D}$, the algebra of distal functions. We also show that the family $\mathcal{S}^d$ of strictly ergodic functions in $\mathcal{D}$ does not form an algebra and hence in particular does not coincide with $\mathcal{W}$. We then use similar constructions to show that a function which is a multiplier for strict ergodicity, either within $\mathcal{D}$ or in general, is necessarily a constant. An example of a metric, strictly ergodic, distal flow is constructed which admits a non-strictly ergodic $2$-fold minimal self-joining. It then follows that the enveloping group of this flow is not strictly ergodic (as a $T$-flow). Finally we show that the distal, strictly ergodic Heisenberg nil-flow is relatively disjoint over its largest equicontinuous factor from $|\mathcal{W}|$.

It is generally believed that robust training of extremely large networks is critical to their success in real-world applications. However, when taken to the extreme, methods that promote robustness can hurt the model's sensitivity to rare or underrepresented patterns. In this paper, we discuss this trade-off between sensitivity and robustness to natural (non-adversarial) perturbations by introducing two notions: contextual feature utility and contextual feature sensitivity. We propose Feature Contrastive Learning (FCL) that encourages a model to be more sensitive to the features that have higher contextual utility. Empirical results demonstrate that models trained with FCL achieve a better balance of robustness and sensitivity, leading to improved generalization in the presence of noise on both vision and NLP datasets., Comment: 31 pages, 5 figures, 3 tables

Deep learning has yielded extraordinary results in vision and natural language processing, but this achievement comes at a cost. Most models require enormous resources during training, both in terms of computation and in human labeling effort. We show that we can identify informative and diverse subsets of data that lead to deep learning models with similar performance as the ones trained with the original dataset. Prior methods have exploited diversity and uncertainty in submodular objective functions for choosing subsets. In addition to these measures, we show that balancing constraints on predicted class labels and decision boundaries are beneficial. We propose a novel formulation of these constraints using matroids, an algebraic structure that generalizes linear independence in vector spaces, and present an efficient greedy algorithm with constant approximation guarantees. We outperform competing baselines on standard classification datasets such as CIFAR-10, CIFAR-100, ImageNet, as well as long-tailed datasets such as CIFAR-100-LT., Comment: Added error bars to the experiments

Deep Convolutional Neural Networks (CNNs) have long been the architecture of choice for computer vision tasks. Recently, Transformer-based architectures like Vision Transformer (ViT) have matched or even surpassed ResNets for image classification. However, details of the Transformer architecture -- such as the use of non-overlapping patches -- lead one to wonder whether these networks are as robust. In this paper, we perform an extensive study of a variety of different measures of robustness of ViT models and compare the findings to ResNet baselines. We investigate robustness to input perturbations as well as robustness to model perturbations. We find that when pre-trained with a sufficient amount of data, ViT models are at least as robust as the ResNet counterparts on a broad range of perturbations. We also find that Transformers are robust to the removal of almost any single layer, and that while activations from later layers are highly correlated with each other, they nevertheless play an important role in classification., Comment: Accepted for publication at ICCV 2021. Rewrote Section 5 and made other minor changes throughout

Todorcevi\'{c}' trichotomy in the class of separable Rosenthal compacta induces a hierarchy in the class of tame (compact, metrizable) dynamical systems $(X,T)$ according to the topological properties of their enveloping semigroups $E(X)$. More precisely, we define the classes $\mathrm{Tame}_\mathbf{2} \subset \mathrm{Tame}_\mathbf{1} \subset \mathrm{Tame},$ where $\mathrm{Tame}_\mathbf{1}$ is the proper subclass of tame systems with first countable $E(X)$, and $\mathrm{Tame}_\mathbf{2}$ is its proper subclass consisting of systems with hereditarily separable $E(X)$. We study some general properties of these classes and exhibit many examples to illustrate these properties., Comment: 37 pages, to appear in Trans. AMS

We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli., Comment: Corrections to the proof of Theorem 6.1

We prove that the maximal infinite step pro-nilfactor $X_\infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $\pi:X \rightarrow X_\infty$, the induced open extension $\pi^*:X^* \rightarrow X^*_\infty$ has the following property: for $x$ in a dense $G_\delta$ set of $X^*$, the orbit closure $L_x=\overline{{\mathcal{O}}}((x,x,\ldots,x), T\times T^2\times \ldots \times T^d)$ is $(\pi^*)^{(d)}$-saturated, i.e. $L_x=((\pi^*)^{(d)})^{-1}(\pi^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $k\ge 2$, then for any $d\in {\mathbb N}$ and any $0\le j

We show the existence, over an arbitrary infinite ergodic $\mathbb{Z}$-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower., Comment: Improved presentation and corrected misprints