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Equipping the rototranslation group $SE(2)$ with a sub-Riemannian structure inspired by the visual cortex V1, we propose algorithms for image inpainting and enhancement based on hypoelliptic diffusion. We innovate on previous implementations of the methods by Citti, Sarti and Boscain et al., by proposing an alternative that prevents fading and capable of producing sharper results in a procedure that we call WaxOn-WaxOff. We also exploit the sub-Riemannian structure to define a completely new unsharp using $SE(2)$, analogous of the classical unsharp filter for 2D image processing, with applications to image enhancement. We demonstrate our method on blood vessels enhancement in retinal scans.

We study the problem of private distribution learning with access to public data. In this setup, which we refer to as public-private learning, the learner is given public and private samples drawn from an unknown distribution $p$ belonging to a class $\mathcal Q$, with the goal of outputting an estimate of $p$ while adhering to privacy constraints (here, pure differential privacy) only with respect to the private samples. We show that the public-private learnability of a class $\mathcal Q$ is connected to the existence of a sample compression scheme for $\mathcal Q$, as well as to an intermediate notion we refer to as list learning. Leveraging this connection: (1) approximately recovers previous results on Gaussians over $\mathbb R^d$; and (2) leads to new ones, including sample complexity upper bounds for arbitrary $k$-mixtures of Gaussians over $\mathbb R^d$, results for agnostic and distribution-shift resistant learners, as well as closure properties for public-private learnability under taking mixtures and products of distributions. Finally, via the connection to list learning, we show that for Gaussians in $\mathbb R^d$, at least $d$ public samples are necessary for private learnability, which is close to the known upper bound of $d+1$ public samples.

Classical simulations are essential for the development of quantum computing, and their exponential scaling can easily fill any modern supercomputer. In this paper we consider the performance and energy consumption of large Quantum Fourier Transform (QFT) simulations run on ARCHER2, the UK's National Supercomputing Service, with QuEST toolkit. We take into account CPU clock frequency and node memory size, and use cache-blocking to rearrange the circuit, which minimises communications. We find that using 2.00GHz instead of 2.25GHz can save as much as 25% of energy at 5% increase in runtime. Higher node memory also has the potential to be more efficient, and cost the user fewer CUs, but at higher runtime penalty. Finally, we present a cache-blocking QFT circuit, which halves the required communication. All our optimisations combined result in 40% faster simulations and 35% energy savings in 44 qubit simulations on 4,096 ARCHER2 nodes.

In the kernelized bandit problem, a learner aims to sequentially compute the optimum of a function lying in a reproducing kernel Hilbert space given only noisy evaluations at sequentially chosen points. In particular, the learner aims to minimize regret, which is a measure of the suboptimality of the choices made. Arguably the most popular algorithm is the Gaussian Process Upper Confidence Bound (GP-UCB) algorithm, which involves acting based on a simple linear estimator of the unknown function. Despite its popularity, existing analyses of GP-UCB give a suboptimal regret rate, which fails to be sublinear for many commonly used kernels such as the Mat\'ern kernel. This has led to a longstanding open question: are existing regret analyses for GP-UCB tight, or can bounds be improved by using more sophisticated analytical techniques? In this work, we resolve this open question and show that GP-UCB enjoys nearly optimal regret. In particular, our results yield sublinear regret rates for the Mat\'ern kernel, improving over the state-of-the-art analyses and partially resolving a COLT open problem posed by Vakili et al. Our improvements rely on a key technical contribution -- regularizing kernel ridge estimators in proportion to the smoothness of the underlying kernel $k$. Applying this key idea together with a largely overlooked concentration result in separable Hilbert spaces (for which we provide an independent, simplified derivation), we are able to provide a tighter analysis of the GP-UCB algorithm.

Past works have shown that the Bernstein-von Mises theorem, on the asymptotic normality of posterior distributions, holds if the parameter dimension $d$ grows slower than the cube root of sample size $n$. Here, we prove the first Bernstein-von Mises theorem in the regime $d^2\ll n$. We establish this result for 1) exponential families and 2) logistic regression with Gaussian design. The proof builds on our recent work on the accuracy of the Laplace approximation to posterior distributions, in which we showed the approximation error in TV distance scales as $d/\sqrt n$.

Programming-by-example (PBE) systems aim to alleviate the burden of programming. However, user-specified examples are often ambiguous, leaving multiple programs to satisfy the specification. Consequently, in most prior work, users have had to provide additional examples, particularly negative ones, to further constrain the search over compatible programs. Recent work resolves additional ambiguity by modeling program synthesis tasks as pragmatic communication, showing promising results on a graphics domain using a rudimentary user-study. We adapt pragmatic reasoning to a sub-domain of regular expressions and rigorously study its usability as a means of communication both with and without the ability to provide negative examples. Our user study (N=30) demonstrates that, with a pragmatic synthesizer, end-users can more successfully communicate a target regex using positive examples alone (95%) compared to using a non-pragmatic synthesizer (51%). Further, users can communicate more efficiently (57% fewer examples) with a pragmatic synthesizer compared to a non-pragmatic one.

Linear Dynamical Systems, both discrete and continuous, are invaluable mathematical models in a plethora of applications such the verification of probabilistic systems, model checking, computational biology, cyber-physical systems, and economics. We consider discrete Linear Recurrence Sequences and continuous C-finite functions, i.e. solutions to homogeneous Linear Differential Equations. The Ultimate Positivity Problem gives the recurrence relation and the initialisation as input and asks whether there is a step $n_0$ (resp. a time $t_0$) such that the Linear Recurrence Sequence $u[n] \ge 0$ for $n > n_0$ (resp. solution to homogeneous linear differential equation $u(t) \ge 0$ for $t > t_0$). There are intrinsic number-theoretic challenges to surmount in order to decide these problems, which crucially arise in engineering and the practical sciences. In these settings, the difficult corner cases are seldom relevant: tolerance to the inherent imprecision is especially critical. We thus characterise \textit{robust} instances of the Ultimate Positivity Problem, i.e.\ inputs for which the decision is locally constant. We describe the sets of Robust YES and Robust NO instances using the First Order Theory of the Reals. We show, via the admission of quantifier elimination by the First Order Theory of the Reals, that these sets are semialgebraic.

Virtual try-on is a critical image synthesis task that aims to transfer clothes from one image to another while preserving the details of both humans and clothes. While many existing methods rely on Generative Adversarial Networks (GANs) to achieve this, flaws can still occur, particularly at high resolutions. Recently, the diffusion model has emerged as a promising alternative for generating high-quality images in various applications. However, simply using clothes as a condition for guiding the diffusion model to inpaint is insufficient to maintain the details of the clothes. To overcome this challenge, we propose an exemplar-based inpainting approach that leverages a warping module to guide the diffusion model's generation effectively. The warping module performs initial processing on the clothes, which helps to preserve the local details of the clothes. We then combine the warped clothes with clothes-agnostic person image and add noise as the input of diffusion model. Additionally, the warped clothes is used as local conditions for each denoising process to ensure that the resulting output retains as much detail as possible. Our approach, namely Diffusion-based Conditional Inpainting for Virtual Try-ON (DCI-VTON), effectively utilizes the power of the diffusion model, and the incorporation of the warping module helps to produce high-quality and realistic virtual try-on results. Experimental results on VITON-HD demonstrate the effectiveness and superiority of our method.

We study the expressive power of first-order logic with counting quantifiers, especially the $k$-variable and quantifier-rank-$q$ fragment $\mathsf{C}^k_q$, using homomorphism indistinguishability. Recently, Dawar, Jakl, and Reggio (2021) proved that two graphs satisfy the same $\mathsf{C}^k_q$-sentences if and only if they are homomorphism indistinguishable over the class $\mathcal{T}^k_q$ of graphs admitting a $k$-pebble forest cover of depth $q$. Their proof builds on the categorical framework of game comonads developed by Abramsky, Dawar, and Wang (2017). We reprove their result using elementary techniques inspired by Dvo\v{r}\'ak (2010). Using these techniques we also give a characterisation of guarded counting logic. Our main focus, however, is to provide a graph theoretic analysis of the graph class $\mathcal{T}^k_q$. This allows us to separate $\mathcal{T}^k_q$ from the intersection of the graph class $\mathcal{TW}_{k-1}$, that is graphs of treewidth less or equal $k-1$, and $\mathcal{TD}_q$, that is graphs of treedepth at most $q$ if $q$ is sufficiently larger than $k$. We are able to lift this separation to the semantic separation of the respective homomorphism indistinguishability relations. A part of this separation is to prove that the class $\mathcal{TD}_q$ is homomorphism distinguishing closed, which was already conjectured by Roberson (2022).

Regression experts consistently recommend plotting residuals for model diagnosis, despite the availability of many numerical hypothesis test procedures designed to use residuals to assess problems with a model fit. Here we provide evidence for why this is good advice using data from a visual inference experiment. We show how conventional tests are too sensitive, which means that too often the conclusion would be that the model fit is inadequate. The experiment uses the lineup protocol which puts a residual plot in the context of null plots. This helps generate reliable and consistent reading of residual plots for better model diagnosis. It can also help in an obverse situation where a conventional test would fail to detect a problem with a model due to contaminated data. The lineup protocol also detects a range of departures from good residuals simultaneously.