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Comparative analysis of scalar fields in scientific visualization often involves distance functions on topological abstractions. This paper focuses on the merge tree abstraction (representing the nesting of sub- or superlevel sets) and proposes the application of the unconstrained deformation-based edit distance. Previous approaches on merge trees often suffer from instability: small perturbations in the data can lead to large distances of the abstractions. While some existing methods can handle so-called vertical instability, the unconstrained deformation-based edit distance addresses both vertical and horizontal instabilities, also called saddle swaps. We establish the computational complexity as NP-complete, and provide an integer linear program formulation for computation. Experimental results on the TOSCA shape matching ensemble provide evidence for the stability of the proposed distance. We thereby showcase the potential of handling saddle swaps for comparison of scalar fields through merge trees.

We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and quantum circuits of shorter depths. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones. Additionally, using this method, we establish a relationship between discrete-time and continuous-time quantum walks, making inroads into a long-standing open problem.

We give a quantum approximation scheme (i.e., $(1 + \varepsilon)$-approximation for every $\varepsilon > 0$) for the classical $k$-means clustering problem in the QRAM model with a running time that has only polylogarithmic dependence on the number of data points. More specifically, given a dataset $V$ with $N$ points in $\mathbb{R}^d$ stored in QRAM data structure, our quantum algorithm runs in time $\tilde{O} \left( 2^{\tilde{O}(\frac{k}{\varepsilon})} \eta^2 d\right)$ and with high probability outputs a set $C$ of $k$ centers such that $cost(V, C) \leq (1+\varepsilon) \cdot cost(V, C_{OPT})$. Here $C_{OPT}$ denotes the optimal $k$-centers, $cost(.)$ denotes the standard $k$-means cost function (i.e., the sum of the squared distance of points to the closest center), and $\eta$ is the aspect ratio (i.e., the ratio of maximum distance to minimum distance). This is the first quantum algorithm with a polylogarithmic running time that gives a provable approximation guarantee of $(1+\varepsilon)$ for the $k$-means problem. Also, unlike previous works on unsupervised learning, our quantum algorithm does not require quantum linear algebra subroutines and has a running time independent of parameters (e.g., condition number) that appear in such procedures.

We consider the problem of estimating a nested structure of two expectations taking the form $U_0 = E[\max\{U_1(Y), \pi(Y)\}]$, where $U_1(Y) = E[X\ |\ Y]$. Terms of this form arise in financial risk estimation and option pricing. When $U_1(Y)$ requires approximation, but exact samples of $X$ and $Y$ are available, an antithetic multilevel Monte Carlo (MLMC) approach has been well-studied in the literature. Under general conditions, the antithetic MLMC estimator obtains a root mean squared error $\varepsilon$ with order $\varepsilon^{-2}$ cost. If, additionally, $X$ and $Y$ require approximate sampling, careful balancing of the various aspects of approximation is required to avoid a significant computational burden. Under strong convergence criteria on approximations to $X$ and $Y$, randomised multilevel Monte Carlo techniques can be used to construct unbiased Monte Carlo estimates of $U_1$, which can be paired with an antithetic MLMC estimate of $U_0$ to recover order $\varepsilon^{-2}$ computational cost. In this work, we instead consider biased multilevel approximations of $U_1(Y)$, which require less strict assumptions on the approximate samples of $X$. Extensions to the method consider an approximate and antithetic sampling of $Y$. Analysis shows the resulting estimator has order $\varepsilon^{-2}$ asymptotic cost under the conditions required by randomised MLMC and order $\varepsilon^{-2}|\log\varepsilon|^3$ cost under more general assumptions.

Chain-of-Though (CoT) prompting has shown promising performance in various reasoning tasks. Recently, Self-Consistency \citep{wang2023selfconsistency} proposes to sample a diverse set of reasoning chains which may lead to different answers while the answer that receives the most votes is selected. In this paper, we propose a novel method to use backward reasoning in verifying candidate answers. We mask a token in the question by ${\bf x}$ and ask the LLM to predict the masked token when a candidate answer is provided by \textit{a simple template}, i.e., ``\textit{\textbf{If we know the answer of the above question is \{a candidate answer\}, what is the value of unknown variable ${\bf x}$?}}'' Intuitively, the LLM is expected to predict the masked token successfully if the provided candidate answer is correct. We further propose FOBAR to combine forward and backward reasoning for estimating the probability of candidate answers. We conduct extensive experiments on six data sets and three LLMs. Experimental results demonstrate that FOBAR achieves state-of-the-art performance on various reasoning benchmarks.

Let $X$ be a set of items of size $n$ that contains some defective items, denoted by $I$, where $I \subseteq X$. In group testing, a {\it test} refers to a subset of items $Q \subset X$. The outcome of a test is $1$ if $Q$ contains at least one defective item, i.e., $Q\cap I \neq \emptyset$, and $0$ otherwise. We give a novel approach to obtaining lower bounds in non-adaptive randomized group testing. The technique produced lower bounds that are within a factor of $1/{\log\log\stackrel{k}{\cdots}\log n}$ of the existing upper bounds for any constant~$k$. Employing this new method, we can prove the following result. For any fixed constants $k$, any non-adaptive randomized algorithm that, for any set of defective items $I$, with probability at least $2/3$, returns an estimate of the number of defective items $|I|$ to within a constant factor requires at least $$\Omega\left(\frac{\log n}{\log\log\stackrel{k}{\cdots}\log n}\right)$$ tests. Our result almost matches the upper bound of $O(\log n)$ and solves the open problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math., Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of $\Omega(\log n/\log\log n)$ previously established by Bshouty [ISAAC 2019].

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 present Self-Driven Strategy Learning ($\textit{sdsl}$), a lightweight online learning methodology for automated reasoning tasks that involve solving a set of related problems. $\textit{sdsl}$ does not require offline training, but instead automatically constructs a dataset while solving earlier problems. It fits a machine learning model to this data which is then used to adjust the solving strategy for later problems. We formally define the approach as a set of abstract transition rules. We describe a concrete instance of the sdsl calculus which uses conditional sampling for generating data and random forests as the underlying machine learning model. We implement the approach on top of the Kissat solver and show that the combination of Kissat+$\textit{sdsl}$ certifies larger bounds and finds more counter-examples than other state-of-the-art bounded model checking approaches on benchmarks obtained from the latest Hardware Model Checking Competition.

In Linear Logic ($\mathsf{LL}$), the exponential modality $!$ brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic ($\mathsf{DiLL}$) is an extension of Linear Logic which includes additional rules for $!$ which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of $\mathsf{GDiLL}$.

Cross-view geolocalization, a supplement or replacement for GPS, localizes an agent within a search area by matching ground-view images to overhead images. Significant progress has been made assuming a panoramic ground camera. Panoramic cameras' high complexity and cost make non-panoramic cameras more widely applicable, but also more challenging since they yield less scene overlap between ground and overhead images. This paper presents Restricted FOV Wide-Area Geolocalization (ReWAG), a cross-view geolocalization approach that combines a neural network and particle filter to globally localize a mobile agent with only odometry and a non-panoramic camera. ReWAG creates pose-aware embeddings and provides a strategy to incorporate particle pose into the Siamese network, improving localization accuracy by a factor of 100 compared to a vision transformer baseline. This extended work also presents ReWAG*, which improves upon ReWAG's generalization ability in previously unseen environments. ReWAG* repeatedly converges accurately on a dataset of images we have collected in Boston with a 72 degree field of view (FOV) camera, a location and FOV that ReWAG* was not trained on.