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Since the work of Polyanskiy, Poor and Verd\'u on the finite blocklength performance of capacity-achieving codes for discrete memoryless channels, many papers have attempted to find further results for more practically relevant channels. However, it seems that the complexity of computing capacity-achieving codes has not been investigated until now. We study this question for the simplest non-trivial Gaussian channels, i.e., the additive colored Gaussian noise channel. To assess the computational complexity, we consider the classes $\mathrm{FP}_1$ and $\#\mathrm{P}_1$. $\mathrm{FP}_1$ includes functions computable by a deterministic Turing machine in polynomial time, whereas $\#\mathrm{P}_1$ encompasses functions that count the number of solutions verifiable in polynomial time. It is widely assumed that $\mathrm{FP}_1\neq\#\mathrm{P}_1$. It is of interest to determine the conditions under which, for a given $M \in \mathbb{N}$, where $M$ describes the precision of the deviation of $C(P,N)$, for a certain blocklength $n_M$ and a decoding error $\epsilon > 0$ with $\epsilon\in\mathbb{Q}$, the following holds: $R_{n_M}(\epsilon)>C(P,N)-\frac{1}{2^M}$. It is shown that there is a polynomial-time computable $N_*$ such that for sufficiently large $P_*\in\mathbb{Q}$, the sequences $\{R_{n_M}(\epsilon)\}_{{n_M}\in\mathbb{N}}$, where each $R_{n_M}(\epsilon)$ satisfies the previous condition, cannot be computed in polynomial time if $\mathrm{FP}_1\neq\#\mathrm{P}_1$. Hence, the complexity of computing the sequence $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ grows faster than any polynomial as $M$ increases. Consequently, it is shown that either the sequence of achievable rates $\{R_{n_M}(\epsilon)\}_{n_M\in\mathbb{N}}$ as a function of the blocklength, or the sequence of blocklengths $\{n_M\}_{M\in\mathbb{N}}$ corresponding to the achievable rates, is not a polynomial-time computable sequence.

Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and the complement of this event is called a failure, then such a random projection likely results in no failures. Assuming a Gaussian random projection, the lemma is proved by showing that the no-failure probability is positive using a combination of Bonferroni's inequality and Markov's inequality. This paper modifies this proof in two ways to obtain a greater lower bound on the no-failure probability. First, Bonferroni's inequality is applied to pairs of failures instead of individual failures. Second, since a pair of projection errors has a bivariate gamma distribution, the probability of a pair of successes is bounded using an inequality from Jensen (1969). If $n$ is the number of points to be embedded and $\mu$ is the probability of a success, then this leads to an increase in the lower bound on the no-failure probability of $\frac{1}{2}\binom{n}{2}(1-\mu)^2$ if $\binom{n}{2}$ is even and $\frac{1}{2}\left(\binom{n}{2}-1\right)(1-\mu)^2$ if $\binom{n}{2}$ is odd. For example, if $n=10^5$ points are to be embedded in $k=10^4$ dimensions with a tolerance of $\epsilon=0.1$, then the improvement in the lower bound is on the order of $10^{-14}$. We also show that further improvement is possible if the inequality in Jensen (1969) extends to three successes, though we do not have a proof of this result.

Deep neural networks and other modern machine learning models are often susceptible to adversarial attacks. Indeed, an adversary may often be able to change a model's prediction through a small, directed perturbation of the model's input - an issue in safety-critical applications. Adversarially robust machine learning is usually based on a minmax optimisation problem that minimises the machine learning loss under maximisation-based adversarial attacks. In this work, we study adversaries that determine their attack using a Bayesian statistical approach rather than maximisation. The resulting Bayesian adversarial robustness problem is a relaxation of the usual minmax problem. To solve this problem, we propose Abram - a continuous-time particle system that shall approximate the gradient flow corresponding to the underlying learning problem. We show that Abram approximates a McKean-Vlasov process and justify the use of Abram by giving assumptions under which the McKean-Vlasov process finds the minimiser of the Bayesian adversarial robustness problem. We discuss two ways to discretise Abram and show its suitability in benchmark adversarial deep learning experiments.

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.

Large language models with vision capabilities (VLMs), e.g., GPT-4o and Gemini 1.5 Pro are powering countless image-text applications and scoring high on many vision-understanding benchmarks. We propose BlindTest, a suite of 7 visual tasks absurdly easy to humans such as identifying (a) whether two circles overlap; (b) whether two lines intersect; (c) which letter is being circled in a word; and (d) counting the number of circles in a Olympic-like logo. Surprisingly, four state-of-the-art VLMs are, on average, only 56.20% accurate on our benchmark, with \newsonnet being the best (73.77% accuracy). On BlindTest, VLMs struggle with tasks that requires precise spatial information and counting (from 0 to 10), sometimes providing an impression of a person with myopia seeing fine details as blurry and making educated guesses. Code is available at: //vlmsareblind.github.io/

The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and $J$-Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators.

A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis.

Mutual information has many applications in image alignment and matching, mainly due to its ability to measure the statistical dependence between two images, even if the two images are from different modalities (e.g., CT and MRI). It considers not only the pixel intensities of the images but also the spatial relationships between the pixels. In this project, we apply the mutual information formula to image matching, where image A is the moving object and image B is the target object and calculate the mutual information between them to evaluate the similarity between the images. For comparison, we also used entropy and information-gain methods to test the dependency of the images. We also investigated the effect of different environments on the mutual information of the same image and used experiments and plots to demonstrate.

We prove the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation. The boundary condition is enforced through the use of the notion of asymptotic cone. The differential operator is discretized based on a discrete analogue of the subdifferential.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.