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Quantum computing has recently emerged as a transformative technology. Yet, its promised advantages rely on efficiently translating quantum operations into viable physical realizations. In this work, we use generative machine learning models, specifically denoising diffusion models (DMs), to facilitate this transformation. Leveraging text-conditioning, we steer the model to produce desired quantum operations within gate-based quantum circuits. Notably, DMs allow to sidestep during training the exponential overhead inherent in the classical simulation of quantum dynamics -- a consistent bottleneck in preceding ML techniques. We demonstrate the model's capabilities across two tasks: entanglement generation and unitary compilation. The model excels at generating new circuits and supports typical DM extensions such as masking and editing to, for instance, align the circuit generation to the constraints of the targeted quantum device. Given their flexibility and generalization abilities, we envision DMs as pivotal in quantum circuit synthesis, enhancing both practical applications but also insights into theoretical quantum computation.

Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints are thrice differentiable, and have Lipschitz first and second derivatives on the feasible region. We provide an IPM that, starting from a strictly feasible point, finds a $\mu$-approximate Fritz John point by solving $\mathcal{O}( \mu^{-7/4})$ trust-region subproblems. For IPMs that handle nonlinear constraints, this result represents the first iteration bound with a polynomial dependence on $1/\mu$. We also show how to use our method to find scaled-KKT points starting from an infeasible solution and improve on existing complexity bounds.

We study two classic variants of block-structured integer programming. Two-stage stochastic programs are integer programs of the form $\{A_i \mathbf{x} + D_i \mathbf{y}_i = \mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where $A_i$ and $D_i$ are bounded-size matrices. On the other hand, $n$-fold programs are integer programs of the form $\{{\sum_{i=1}^n C_i\mathbf{y}_i=\mathbf{a}} \textrm{ and } D_i\mathbf{y}_i=\mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where again $C_i$ and $D_i$ are bounded-size matrices. It is known that solving these kind of programs is fixed-parameter tractable when parameterized by the maximum dimension among the relevant matrices $A_i,C_i,D_i$ and the maximum absolute value of any entry appearing in the constraint matrix. We show that the parameterized tractability results for two-stage stochastic and $n$-fold programs persist even when one allows large entries in the global part of the program. More precisely, we prove that: - The feasibility problem for two-stage stochastic programs is fixed-parameter tractable when parameterized by the dimensions of matrices $A_i,D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we allow matrices $A_i$ to have arbitrarily large entries. - The linear optimization problem for $n$-fold integer programs that are uniform -- all matrices $C_i$ are equal -- is fixed-parameter tractable when parameterized by the dimensions of matrices $C_i$ and $D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we require that $C_i=C$ for all $i=1,\ldots,n$, but we allow $C$ to have arbitrarily large entries. In the second result, the uniformity assumption is necessary; otherwise the problem is $\mathsf{NP}$-hard already when the parameters take constant values. Both our algorithms are weakly polynomial: the running time is measured in the total bitsize of the input.

We establish conditions under which latent causal graphs are nonparametrically identifiable and can be reconstructed from unknown interventions in the latent space. Our primary focus is the identification of the latent structure in measurement models without parametric assumptions such as linearity or Gaussianity. Moreover, we do not assume the number of hidden variables is known, and we show that at most one unknown intervention per hidden variable is needed. This extends a recent line of work on learning causal representations from observations and interventions. The proofs are constructive and introduce two new graphical concepts -- imaginary subsets and isolated edges -- that may be useful in their own right. As a matter of independent interest, the proofs also involve a novel characterization of the limits of edge orientations within the equivalence class of DAGs induced by unknown interventions. These are the first results to characterize the conditions under which causal representations are identifiable without making any parametric assumptions in a general setting with unknown interventions and without faithfulness.

We propose a learning-based compression scheme that envelopes a standard codec between pre and post-processing deep CNNs. Specifically, we demonstrate improvements over prior approaches utilizing a compression-decompression network by introducing: (a) an edge-aware loss function to prevent blurring that is commonly occurred in prior works & (b) a super-resolution convolutional neural network (CNN) for post-processing along with a corresponding pre-processing network for improved rate-distortion performance in the low rate regime. The algorithm is assessed on a variety of datasets varying from low to high resolution namely Set 5, Set 7, Classic 5, Set 14, Live 1, Kodak, General 100, CLIC 2019. When compared to JPEG, JPEG2000, BPG, and recent CNN approach, the proposed algorithm contributes significant improvement in PSNR with an approximate gain of 20.75%, 8.47%, 3.22%, 3.23% and 24.59%, 14.46%, 10.14%, 8.57% at low and high bit-rates respectively. Similarly, this improvement in MS-SSIM is approximately 71.43%, 50%, 36.36%, 23.08%, 64.70% and 64.47%, 61.29%, 47.06%, 51.52%, 16.28% at low and high bit-rates respectively. With CLIC 2019 dataset, PSNR is found to be superior with approximately 16.67%, 10.53%, 6.78%, and 24.62%, 17.39%, 14.08% at low and high bit-rates respectively, over JPEG2000, BPG, and recent CNN approach. Similarly, the MS-SSIM is found to be superior with approximately 72%, 45.45%, 39.13%, 18.52%, and 71.43%, 50%, 41.18%, 17.07% at low and high bit-rates respectively, compared to the same approaches. A similar type of improvement is achieved with other datasets also.

We introduce PUNQ, a novel quantum programming language with quantum control, which features higher-order programs that can be superposed, enabling quantum control via quantum conditionals. Our language boasts a type system guaranteeing both unitarity and polynomial-time normalization. Unitarity is achieved by using a special modality for superpositions while requiring orthogonality among superposed terms. Polynomial-time normalization is achieved using a linear-logic-based type discipline employing Barber and Plotkin duality along with a specific modality to account for potential duplications. This type discipline also guarantees that derived values have polynomial size. PUNQ seamlessly combines the two modalities: quantum circuit programs uphold unitarity, and all programs are evaluated in polynomial time, ensuring their feasibility.

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the geometry of the domain.

We propose a new variable selection procedure for a functional linear model with multiple scalar responses and multiple functional predictors. This method is based on basis expansions of the involved functional predictors and coefficients that lead to a multivariate linear regression model. Then a criterion by means of which the variable selection problem reduces to that of estimating a suitable set is introduced. Estimation of this set is achieved by using appropriate penalizations of estimates of this criterion, so leading to our proposal. A simulation study that permits to investigate the effectiveness of the proposed approach and to compare it with existing methods is given.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.