The asymptotically optimal version of Rapidly-exploring Random Tree (RRT*) is often used to find optimal paths in a high-dimensional configuration space. The well-known issue of RRT* is its slow convergence towards the optimal solution. A possible solution is to draw random samples only from a subset of the configuration space that is known to contain configurations that can improve the cost of the path (omniscient set). A fast convergence rate may be achieved by approximating the omniscient with a low-volume set. In this letter, we propose new methods to approximate the omniscient set and methods for their effective sampling. First, we propose to approximate the omniscient set using several (small) hyperellipsoids defined by sections of the current best solution. The second approach approximates the omniscient set by a convex hull computed from the current solution. Both approaches ensure asymptotical optimality and work in a general n-dimensional configuration space. The experiments have shown superior performance of our approaches in multiple scenarios in 3D and 6D configuration spaces.

Music source separation is an audio-to-audio retrieval task of extracting one or more constituent components, or composites thereof, from a musical audio mixture. Each of these constituent components is often referred to as a "stem" in literature. Historically, music source separation has been dominated by a stem-based paradigm, leading to most state-of-the-art systems being either a collection of single-stem extraction models, or a tightly coupled system with a fixed, difficult-to-modify, set of supported stems. Combined with the limited data availability, advances in music source separation have thus been mostly limited to the "VDBO" set of stems: \textit{vocals}, \textit{drum}, \textit{bass}, and the catch-all \textit{others}. Recent work in music source separation has begun to challenge the fixed-stem paradigm, moving towards models able to extract any musical sound as long as this target type of sound could be specified to the model as an additional query input. We generalize this idea to a \textit{query-by-region} source separation system, specifying the target based on the query regardless of how many sound sources or which sound classes are contained within it. To do so, we propose the use of hyperellipsoidal regions as queries to allow for an intuitive yet easily parametrizable approach to specifying both the target (location) as well as its spread. Evaluation of the proposed system on the MoisesDB dataset demonstrated state-of-the-art performance of the proposed system both in terms of signal-to-noise ratios and retrieval metrics.

A technical note aiming to offer deeper intuition for the LayerNorm function common in deep neural networks. LayerNorm is defined relative to a distinguished 'neural' basis, but it does more than just normalize the corresponding vector elements. Rather, it implements a composition -- of linear projection, nonlinear scaling, and then affine transformation -- on input activation vectors. We develop both a new mathematical expression and geometric intuition, to make the net effect more transparent. We emphasize that, when LayerNorm acts on an N-dimensional vector space, all outcomes of LayerNorm lie within the intersection of an (N-1)-dimensional hyperplane and the interior of an N-dimensional hyperellipsoid. This intersection is the interior of an (N-1)-dimensional hyperellipsoid, and typical inputs are mapped near its surface. We find the direction and length of the principal axes of this (N-1)-dimensional hyperellipsoid via the eigen-decomposition of a simply constructed matrix.

Sampling-based motion planning algorithms are very effective at finding solutions in high-dimensional continuous state spaces as they do not require prior approximations of the problem domain compared to traditional discrete graph-based searches. The anytime version of the Rapidly-exploring Random Trees (RRT) algorithm, denoted as RRT*, often finds high-quality solutions by incrementally approximating and searching the problem domain through random sampling. However, due to its low sampling efficiency and slow convergence rate, research has proposed many variants of RRT*, incorporating different heuristics and sampling strategies to overcome the constraints in complex planning problems. Yet, these approaches address specific convergence aspects of RRT* limitations, leaving a need for a sampling-based algorithm that can quickly find better solutions in complex high-dimensional state spaces with a faster convergence rate for practical motion planning applications. This article unifies and leverages the greedy search and heuristic techniques used in various RRT* variants to develop a greedy version of the anytime Rapidly-exploring Random Trees algorithm, denoted as Greedy RRT* (G-RRT*). It improves the initial solution-finding time of RRT* by maintaining two trees rooted at both the start and goal ends, advancing toward each other using greedy connection heuristics. It also accelerates the convergence rate of RRT* by introducing a greedy version of direct informed sampling procedure, which guides the sampling towards the promising region of the problem domain based on heuristics. We validate our approach on simulated planning problems, manipulation problems on Barrett WAM Arms, and on a self-reconfigurable robot, Panthera. Results show that G-RRT* produces asymptotically optimal solution paths and outperforms state-of-the-art RRT* variants, especially in high-dimensional planning problems.

In modern approaches to path planning and robot motion planning, anytime almost-surely asymptotically optimal planners dominate the benchmark of sample-based planners. A notable example is Batch Informed Trees (BIT*), where planners iteratively determine paths to groups of vertices within the exploration area. However, maintaining a consistent batch size is crucial for initial pathfinding and optimal performance, relying on effective task allocation. This paper introduces Flexible Informed Tree (FIT*), a novel planner integrating an adaptive batch-size method to enhance task scheduling in various environments. FIT* employs a flexible approach in adjusting batch sizes dynamically based on the inherent complexity of the planning domain and the current n-dimensional hyperellipsoid of the system. By constantly optimizing batch sizes, FIT* achieves improved computational efficiency and scalability while maintaining solution quality. This adaptive batch-size method significantly enhances the planner's ability to handle diverse and evolving problem domains. FIT* outperforms existing single-query, sampling-based planners on the tested problems in R^2 to R^8, and was demonstrated in real-world environments with KI-Fabrik/DARKO-Project Europe.

Local differential privacy (LDP) can be adopted to anonymize richer user data attributes that will be input to sophisticated machine learning (ML) tasks. However, today's LDP approaches are largely task-agnostic and often lead to severe performance loss -- they simply inject noise to all data attributes according to a given privacy budget, regardless of what features are most relevant for the ultimate task. In this paper, we address how to significantly improve the ultimate task performance with multi-dimensional user data by considering a task-aware privacy preservation problem. The key idea is to use an encoder-decoder framework to learn (and anonymize) a task-relevant latent representation of user data. We obtain an analytical near-optimal solution for the linear setting with mean-squared error (MSE) task loss. We also provide an approximate solution through a gradient-based learning algorithm for general nonlinear cases. Extensive experiments demonstrate that our task-aware approach significantly improves ultimate task accuracy compared to standard benchmark LDP approaches with the same level of privacy guarantee.