Dimensionality reduction is a fundamental task in modern data science. Several projection methods specifically tailored to take into account the non-linearity of the data via local embeddings have been proposed. Such methods are often based on local neighbourhood structures and require tuning the number of neighbours that define this local structure, and the dimensionality of the lower-dimensional space onto which the data are projected. Such choices critically influence the quality of the resulting embedding. In this paper, we exploit a recently proposed intrinsic dimension estimator which also returns the optimal locally adaptive neighbourhood sizes according to some desirable criteria. In principle, this adaptive framework can be employed to perform an optimal hyper-parameter tuning of any dimensionality reduction algorithm that relies on local neighbourhood structures. Numerical experiments on both real-world and simulated datasets show that the proposed method can be used to significantly improve well-known projection methods when employed for various learning tasks, with improvements measurable through both quantitative metrics and the quality of low-dimensional visualizations.

Evaluation of per-sample uncertainty quantification from neural networks is essential for decision-making involving high-risk applications. A common approach is to use the predictive distribution from Bayesian or approximation models and decompose the corresponding predictive uncertainty into epistemic (model-related) and aleatoric (data-related) components. However, additive decomposition has recently been questioned. In this work, we propose an intuitive framework for uncertainty estimation and decomposition based on the signal-to-noise ratio of class probability distributions across different model predictions. We introduce a variance-gated measure that scales predictions by a confidence factor derived from ensembles. We use this measure to discuss the existence of a collapse in the diversity of committee machines.

The rise of parallel computing hardware has made it increasingly important to understand which nonlinear state space models can be efficiently parallelized. Recent advances like DEER (arXiv:2309.12252) or DeepPCR (arXiv:2309.16318) have shown that evaluating a state space model can be recast as solving a parallelizable optimization problem, and sometimes this approach can yield dramatic speed-ups in evaluation time. However, the factors that govern the difficulty of these optimization problems remain unclear, limiting the larger adoption of the technique. In this work, we establish a precise relationship between the dynamics of a nonlinear system and the conditioning of its corresponding optimization formulation. We show that the predictability of a system, defined as the degree to which small perturbations in state influence future behavior, impacts the number of optimization steps required for evaluation. In predictable systems, the state trajectory can be computed in $O((\log T)^2)$ time, where $T$ is the sequence length, a major improvement over the conventional sequential approach. In contrast, chaotic or unpredictable systems exhibit poor conditioning, with the consequence that parallel evaluation converges too slowly to be useful. Importantly, our theoretical analysis demonstrates that for predictable systems, the optimization problem is always well-conditioned, whereas for unpredictable systems, the conditioning degrades exponentially as a function of the sequence length. We validate our claims through extensive experiments, providing practical guidance on when nonlinear dynamical systems can be efficiently parallelized, and highlighting predictability as a key design principle for parallelizable models.

Ali Goli 1, Mahdieh Alizadeh 1, and Hadi Sadoghi Yazdi 1,2 1 Department of Computer Engineering, Ferdowsi University of Mashhad, Mashhad, Iran 2 Center of Excellence in Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, Iran April 10, 2025 Abstract Manifold learning techniques, such as Locally linear embedding (LLE), are designed to preserve the local neighborhood structures of high-dimensional data during dimensionality reduction. Traditional LLE employs Euclidean distance to define neighborhoods, which can struggle to capture the intrinsic geometric relationships within complex data. A novel approach, Adaptive locally linear embedding(ALLE), is introduced to address this limitation by incorporating a dynamic, data-driven metric that enhances topological preservation. This method redefines the concept of proximity by focusing on topological neighborhood inclusion rather than fixed distances. By adapting the metric based on the local structure of the data, it achieves superior neighborhood preservation, particularly for datasets with complex geometries and high-dimensional structures. Experimental results demonstrate that ALLE significantly improves the alignment between neighborhoods in the input and feature spaces, resulting in more accurate and topologically faithful embeddings. Keywords-- Manifold Learning, Adaptive Locally Linear Embedding, Dimensionality Reduction, Topological Preservation, Complex Geometries, High-Dimensional Data, Topological Neighborhood Inclusion, Intrinsic Geometric Relationships 1 Introduction Locally linear embedding(LLE) is a prominent manifold learning technique designed to reduce the dimensionality of high-dimensional datasets while preserving their intrinsic geometric structure. Proposed by Roweis and Saul, LLE operates through a systematic process that includes identifying the K-nearest neighbors for each data point, calculating reconstruction weights to express each point as a linear combination of its neighbors, and ultimately generating a low-dimensional representation that retains local relationships [14]. However, LLE traditionally relies on fixed distance metrics, such as Euclidean distance, which may inadequately represent complex data distributions and fail to capture nuanced topological relationships. In response to these limitations, we introduce a novel approach termed Adaptive LLE(ALLE), which integrates a flexible, data-driven metric into the LLE framework.

Artificial Intelligence (AI) song generation has emerged as a popular topic, yet the focus on exploring the latent correlations between specific lyrical and rhythmic features remains limited. In contrast, this pilot study particularly investigates the relationships between keywords and rhythmically stressed features such as strong beats in songs. It focuses on several key elements: keywords or non-keywords, stressed or unstressed syllables, and strong or weak beats, with the aim of uncovering insightful correlations. Experimental results indicate that, on average, 80.8\% of keywords land on strong beats, whereas 62\% of non-keywords fall on weak beats. The relationship between stressed syllables and strong or weak beats is weak, revealing that keywords have the strongest relationships with strong beats. Additionally, the lyrics-rhythm matching score, a key matching metric measuring keywords on strong beats and non-keywords on weak beats across various time signatures, is 0.765, while the matching score for syllable types is 0.495. This study demonstrates that word types strongly align with their corresponding beat types, as evidenced by the distinct patterns, whereas syllable types exhibit a much weaker alignment. This disparity underscores the greater reliability of word types in capturing rhythmic structures in music, highlighting their crucial role in effective rhythmic matching and analysis. We also conclude that keywords that consistently align with strong beats are more reliable indicators of lyrics-rhythm associations, providing valuable insights for AI-driven song generation through enhanced structural analysis. Furthermore, our development of tailored Lyrics-Rhythm Matching (LRM) metrics maximizes lyrical alignments with corresponding beat stresses, and our novel LRM file format captures critical lyrical and rhythmic information without needing original sheet music.

We introduce the Laser Learning Environment (LLE), a collaborative multi-agent reinforcement learning environment in which coordination is central. In LLE, agents depend on each other to make progress (interdependence), must jointly take specific sequences of actions to succeed (perfect coordination), and accomplishing those joint actions does not yield any intermediate reward (zero-incentive dynamics). The challenge of such problems lies in the difficulty of escaping state space bottlenecks caused by interdependence steps since escaping those bottlenecks is not rewarded. We test multiple state-of-the-art value-based MARL algorithms against LLE and show that they consistently fail at the collaborative task because of their inability to escape state space bottlenecks, even though they successfully achieve perfect coordination. We show that Q-learning extensions such as prioritised experience replay and n-steps return hinder exploration in environments with zero-incentive dynamics, and find that intrinsic curiosity with random network distillation is not sufficient to escape those bottlenecks. We demonstrate the need for novel methods to solve this problem and the relevance of LLE as cooperative MARL benchmark.

Several unsupervised learning algorithms based on an eigendecompo- sition provide either an embedding or a clustering only for given train- ing points, with no straightforward extension for out-of-sample examples short of recomputing eigenvectors. This paper provides a unified frame- work for extending Local Linear Embedding (LLE), Isomap, Laplacian Eigenmaps, Multi-Dimensional Scaling (for dimensionality reduction) as well as for Spectral Clustering. This framework is based on seeing these algorithms as learning eigenfunctions of a data-dependent kernel. Numerical experiments show that the generalizations performed have a level of error comparable to the variability of the embedding algorithms due to the choice of training data.

Locally Linear Embedding (LLE) is a nonlinear spectral dimensionality reduction and manifold learning method. It has two main steps which are linear reconstruction and linear embedding of points in the input space and embedding space, respectively. In this work, we look at the linear reconstruction step from a stochastic perspective where it is assumed that every data point is conditioned on its linear reconstruction weights as latent factors. The stochastic linear reconstruction of LLE is solved using expectation maximization. We show that there is a theoretical connection between three fundamental dimensionality reduction methods, i.e., LLE, factor analysis, and probabilistic Principal Component Analysis (PCA). The stochastic linear reconstruction of LLE is formulated similar to the factor analysis and probabilistic PCA. It is also explained why factor analysis and probabilistic PCA are linear and LLE is a nonlinear method. This work combines and makes a bridge between two broad approaches of dimensionality reduction, i.e., the spectral and probabilistic algorithms.

In an earlier article "The Importance of Location in Real Estate, Weather, and Machine Learning," various meanings and applications of location-based discovery in data science and machine learning were discussed. One algorithm described there is a powerful but strangely named machine learning algorithm: the Support Vector Machine (SVM). In the remarks below, we summarize the significance and utility of another powerful but strangely named machine learning algorithm that focuses on location: Local Linear Embedding (LLE). LLE is a specific example from the general category of Manifold Learning algorithms. The most famous example of manifold learning with LLE is the Swiss jelly roll example (illustrated above).

Classification and identification of amino acids in aqueous solutions is important in the study of biomacromolecules. Laser Induced Breakdown Spectroscopy (LIBS) uses high energy laser-pulses for ablation of chemical compounds whose radiated spectra are captured and recorded to reveal molecular structure. Spectral peaks and noise from LIBS are impacted by experimental protocols. Current methods for LIBS spectral analysis achieves promising results using PCA, a linear method. It is well-known that the underlying physical processes behind LIBS are highly nonlinear. Our work set out to understand the impact of LIBS spectra on suitable neighborhood size over which to consider pattern phenomena, if nonlinear methods capture pattern phenomena with increased efficacy, and how they improve classification and identification of compounds. We analyzed four amino acids, polysaccharide, and a control group, water. We developed an information theoretic method for measurement of LIBS energy spectra, implemented manifold methods for nonlinear dimensionality reduction, and found while clustering results were not statistically significantly different, nonlinear methods lead to increased classification accuracy. Moreover, our approach uncovered the contribution of micro-wells (experimental protocol) in LIBS spectra. To the best of our knowledge, ours is the first application of Manifold methods to LIBS amino-acid analysis in the research literature.