In this paper, we demonstrate that the PGW problem can be transformed into a variant of the Gromov-Wasserstein problem, akin to the conversion of the partial optimal transport problem into an optimal transport problem. This transformation leads to two new solvers, mathematically and computationally equivalent, based on the Frank-Wolfe algorithm, that provide efficient solutions to the PGW problem. We further establish that the PGW problem constitutes a metric for metric measure spaces. Finally, we validate the effectiveness of our proposed solvers in terms of computation time and performance on shape-matching and positive-unlabeled learning problems, comparing them against existing baselines.

Computing eigenvalue decomposition (EVD) of a given linear operator, or finding its leading eigenvalues and eigenfunctions, is a fundamental task in many machine learning and scientific computing problems. For high-dimensional eigenvalue problems, training neural networks to parameterize the eigenfunctions is considered as a promising alternative to the classical numerical linear algebra techniques. This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition, accompanied by new techniques called nesting for learning the top-$L$ singular values and singular functions in the correct order. The proposed method promotes the desired orthogonality in the learned functions implicitly and efficiently via an unconstrained optimization formulation, which is easy to solve with off-the-shelf gradient-based optimization algorithms. We demonstrate the effectiveness of the proposed optimization framework for use cases in computational physics and machine learning.

Recent advancements have introduced machine learning frameworks to enhance the Branch and Bound (B\&B) branching policies for solving Mixed Integer Linear Programming (MILP). These methods, primarily relying on imitation learning of Strong Branching, have shown superior performance. However, collecting expert samples for imitation learning, particularly for Strong Branching, is a time-consuming endeavor. To address this challenge, we propose \textbf{C}ontrastive Learning with \textbf{A}ugmented \textbf{M}ILPs for \textbf{Branch}ing (CAMBranch), a framework that generates Augmented MILPs (AMILPs) by applying variable shifting to limited expert data from their original MILPs. This approach enables the acquisition of a considerable number of labeled expert samples. CAMBranch leverages both MILPs and AMILPs for imitation learning and employs contrastive learning to enhance the model's ability to capture MILP features, thereby improving the quality of branching decisions. Experimental results demonstrate that CAMBranch, trained with only 10\% of the complete dataset, exhibits superior performance. Ablation studies further validate the effectiveness of our method.

Probabilistic circuits (PCs) such as sum-product networks efficiently represent large multi-variate probability distributions. They are preferred in practice over other probabilistic representations such as Bayesian and Markov networks because PCs can solve marginal inference (MAR) tasks in time that scales linearly in the size of the network. Unfortunately, the maximum-a-posteriori (MAP) and marginal MAP (MMAP) tasks remain NP-hard in these models. Inspired by the recent work on using neural networks for generating near-optimal solutions to optimization problems such as integer linear programming, we propose an approach that uses neural networks to approximate (M)MAP inference in PCs. The key idea in our approach is to approximate the cost of an assignment to the query variables using a continuous multilinear function, and then use the latter as a loss function. The two main benefits of our new method are that it is self-supervised and after the neural network is learned, it requires only linear time to output a solution. We evaluate our new approach on several benchmark datasets and show that it outperforms three competing linear time approximations, max-product inference, max-marginal inference and sequential estimation, which are used in practice to solve MMAP tasks in PCs.

Locomotion gaits are fundamental for control of soft terrestrial robots. However, synthesis of these gaits is challenging due to modeling of robot-environment interaction and lack of a mathematical framework. This work presents an environment-centric, data-driven and fault-tolerant probabilistic Model-Free Control (pMFC) framework that allows for soft multi-limb robots to learn from their environment and synthesize diverse sets of locomotion gaits for realizing open-loop control. Here, discretization of factors dominating robot-environment interactions enables an environment-specific graphical representation where the edges encode experimental locomotion data corresponding to the robot motion primitives. In this graph, locomotion gaits are defined as simple cycles that are transformation invariant, i.e., the locomotion is independent of the starting vertex of these periodic cycles. Gait synthesis, the problem of finding optimal locomotion gaits for a given substrate, is formulated as Binary Integer Linear Programming (BILP) problems with a linearized cost function, linear constraints, and iterative simple cycle detection. Experimentally, gaits are synthesized for varying robot-environment interactions. Variables include robot morphology - three-limb and four-limb robots, TerreSoRo-III and TerreSoRo-IV; substrate - rubber mat, whiteboard and carpet; and actuator functionality - simulated loss of robot limb actuation. On an average, gait synthesis improves the translation and rotation speeds by 82% and 97% respectively. The results highlight that data-driven methods are vital to soft robot locomotion control due to the significant influence of unexpected asymmetries in the system and the dependence of optimal gait sequences on the experimental robot-environment interaction.

Correlation clustering is a powerful unsupervised learning paradigm that supports positive and negative similarities. In this paper, we assume the similarities are not known in advance. Instead, we employ active learning to iteratively query similarities in a cost-efficient way. In particular, we develop three effective acquisition functions to be used in this setting. One is based on the notion of inconsistency (i.e., when similarities violate the transitive property). The remaining two are based on information-theoretic quantities, i.e., entropy and information gain.

Diffusion models are a class of generative models that function by progressively introducing noise into data and then methodically demonising it [17, 11]. They have revolutionized high-fidelity creation of complex data, and their applications have rapidly expanded beyond mere image synthesis, finding relevance in areas such as engineering [22, 24], automation [3, 13], chemistry [1, 12], and scientific research [2, 6]. Although diffusion models excel at generating content that is coherent and aligns closely with the original data distribution, their direct application in scenarios requiring stringent adherence to predefined criteria poses significant challenges. Particularly in domains where the generated data needs to not only resemble real-world examples but also rigorously comply with established specifications, physical laws, or engineering principles, conventional diffusion models are unable to ensure this level of precision. Given these limitations, one may consider an alternative approach: training a diffusion model on a data distribution that already aligns with these constraints.

The key challenge in the noisy intermediate-scale quantum era is finding useful circuits compatible with current device limitations. Variational quantum algorithms (VQAs) offer a potential solution by fixing the circuit architecture and optimizing individual gate parameters in an external loop. However, parameter optimization can become intractable, and the overall performance of the algorithm depends heavily on the initially chosen circuit architecture. Several quantum architecture search (QAS) algorithms have been developed to design useful circuit architectures automatically. In the case of parameter optimization alone, noise effects have been observed to dramatically influence the performance of the optimizer and final outcomes, which is a key line of study. However, the effects of noise on the architecture search, which could be just as critical, are poorly understood. This work addresses this gap by introducing a curriculum-based reinforcement learning QAS (CRLQAS) algorithm designed to tackle challenges in realistic VQA deployment. The algorithm incorporates (i) a 3D architecture encoding and restrictions on environment dynamics to explore the search space of possible circuits efficiently, (ii) an episode halting scheme to steer the agent to find shorter circuits, and (iii) a novel variant of simultaneous perturbation stochastic approximation as an optimizer for faster convergence. To facilitate studies, we developed an optimized simulator for our algorithm, significantly improving computational efficiency in simulating noisy quantum circuits by employing the Pauli-transfer matrix formalism in the Pauli-Liouville basis. Numerical experiments focusing on quantum chemistry tasks demonstrate that CRLQAS outperforms existing QAS algorithms across several metrics in both noiseless and noisy environments.

Decentralized Federated Learning (DFL) has received significant recent research attention, capturing settings where both model updates and model aggregations -- the two key FL processes -- are conducted by the clients. In this work, we propose Decentralized Sporadic Federated Learning ($\texttt{DSpodFL}$), a DFL methodology which generalizes the notion of sporadicity in both of these processes, modeling the impact of different forms of heterogeneity that manifest in realistic DFL settings. $\texttt{DSpodFL}$ unifies many of the prominent decentralized optimization methods, e.g., distributed gradient descent (DGD), randomized gossip (RG), and decentralized federated averaging (DFedAvg), under a single modeling framework. We analytically characterize the convergence behavior of $\texttt{DSpodFL}$, showing, among other insights, that we can match a geometric convergence rate to a finite optimality gap under more general assumptions than in existing works. Through experiments, we demonstrate that $\texttt{DSpodFL}$ achieves significantly improved training speeds and robustness to variations in system parameters compared to the state-of-the-art.

Recent advances in diffusion models attempt to handle conditional generative tasks by utilizing a differentiable loss function for guidance without the need for additional training. While these methods achieved certain success, they often compromise on sample quality and require small guidance step sizes, leading to longer sampling processes. This paper reveals that the fundamental issue lies in the manifold deviation during the sampling process when loss guidance is employed. We theoretically show the existence of manifold deviation by establishing a certain lower bound for the estimation error of the loss guidance. To mitigate this problem, we propose Diffusion with Spherical Gaussian constraint (DSG), drawing inspiration from the concentration phenomenon in high-dimensional Gaussian distributions. DSG effectively constrains the guidance step within the intermediate data manifold through optimization and enables the use of larger guidance steps. Furthermore, we present a closed-form solution for DSG denoising with the Spherical Gaussian constraint. Notably, DSG can seamlessly integrate as a plugin module within existing training-free conditional diffusion methods. Implementing DSG merely involves a few lines of additional code with almost no extra computational overhead, yet it leads to significant performance improvements. Comprehensive experimental results in various conditional generation tasks validate the superiority and adaptability of DSG in terms of both sample quality and time efficiency.