Bayesian Optimisation (BO) is a powerful tool for optimising expensive blackbox functions but its effectiveness diminishes in highdimensional spaces due to sparse data and poor surrogate model scalability While Variational Autoencoder (VAE) based approaches address this by learning low-dimensional latent representations the reconstructionbased objective function often brings the functional distribution mismatch between the latent space and original space leading to suboptimal optimisation performance In this paper we first analyse the reason why reconstructiononly loss may lead to distribution mismatch and then propose HiBBO a novel BO framework that introduces the space consistency into the latent space construction in VAE using HiPPO - a method for longterm sequence modelling - to reduce the functional distribution mismatch between the latent space and original space Experiments on highdimensional benchmark tasks demonstrate that HiBBO outperforms existing VAEBO methods in convergence speed and solution quality Our work bridges the gap between high-dimensional sequence representation learning and efficient Bayesian Optimisation enabling broader applications in neural architecture search materials science and beyond.

Dimensionality reduction methods such as t-SNE and UMAP are popular methods for visualizing data with a potential (latent) clustered structure. They are known to group data points at the same time as they embed them, resulting in visualizations with well-separated clusters that preserve local information well. However, t-SNE and UMAP also tend to distort the global geometry of the underlying data. We propose a more transparent, modular approach consisting of first clustering the data, then embedding each cluster, and finally aligning the clusters to obtain a global embedding. We demonstrate this approach on several synthetic and real-world datasets and show that it is competitive with existing methods, while being much more transparent.

Building upon the success of low-rank adapter (LoRA), low-rank gradient projection (LoRP) has emerged as a promising solution for memory-efficient fine-tuning. However, existing LoRP methods typically treat each row of the gradient matrix as the default projection unit, leaving the role of projection granularity underexplored. In this work, we propose a novel framework, VLoRP, that extends low-rank gradient projection by introducing an additional degree of freedom for controlling the trade-off between memory efficiency and performance, beyond the rank hyper-parameter. Through this framework, we systematically explore the impact of projection granularity, demonstrating that finer-grained projections lead to enhanced stability and efficiency even under a fixed memory budget. Regarding the optimization for VLoRP, we present ProjFactor, an adaptive memory-efficient optimizer, that significantly reduces memory requirement while ensuring competitive performance, even in the presence of gradient accumulation. Additionally, we provide a theoretical analysis of VLoRP, demonstrating the descent and convergence of its optimization trajectory under both SGD and ProjFactor. Extensive experiments are conducted to validate our findings, covering tasks such as commonsense reasoning, MMLU, and GSM8K.

A NALYTICALD ISCOVERY OF M ANIFOLD WITH M A-CHINE L EARNING Y afei Shen 1 & Huan-Fei Ma 1, & Ling Y ang 1, 1 School of Mathematical Sciences, Soochow University, Suzhou 215006, China A BSTRACT Understanding low-dimensional structures within high-dimensional data is crucial for visualization, interpretation, and denoising in complex datasets. Despite the advancements in manifold learning techniques, key challenges--such as limited global insight and the lack of interpretable analytical descriptions--remain unresolved. In this work, we introduce a novel framework, GAMLA (Global Analytical Manifold Learning using Auto-encoding). GAMLA employs a two-round training process within an auto-encoding framework to derive both character and complementary representations for the underlying manifold. With the character representation, the manifold is represented by a parametric function which unfold the manifold to provide a global coordinate. While with the complementary representation, an approximate explicit manifold description is developed, offering a global and analytical representation of smooth manifolds underlying high-dimensional datasets. This enables the analytical derivation of geometric properties such as curvature and normal vectors. Moreover, we find the two representations together decompose the whole latent space and can thus characterize the local spatial structure surrounding the manifold, proving particularly effective in anomaly detection and categorization. Through extensive experiments on benchmark datasets and real-world applications, GAMLA demonstrates its ability to achieve computational efficiency and interpretability while providing precise geometric and structural insights. This framework bridges the gap between data-driven manifold learning and analytical geometry, presenting a versatile tool for exploring the intrinsic properties of complex data sets. 1 I NTRODUCTION Discovering low-dimensional structures, particularly their geometric properties, from high-dimensional data clouds enables visualization, denoising, and interpretation of complex datasets (Meil a & Zhang, 2023; Belkin & Niyogi, 2003; van der Maaten & Hinton, 2008; McInnes & Healy, 2018; Luo & Hu, 2020). As a result, the concept of manifold learning has attracted significant attention, leading to numerous breakthroughs over the past two decades.

Summary and Contributions: Update: Thanks for addressing the concerns raised by the reviewers, based on re-reading the paper and going over the comments, I am able to understand the experiments better - and based on the authors comments that they will revise the draft to make things more clear, I will change my score to accept. Having said that, I would still keep my confidence low since I am unable to accurately access the significance of the result and I believe that would be a key factor to consider in a novel theoretical paper. The result is based on two key properties of the transformed space that they identify. The first is'safety', which is a measure of how well can we recover the posterior in the original space from the feature space. The second is smoothness, which is a measure of how hard it is to recover the posterior in the original space from the feature space.

We derive a general change of variables formula for score functions, showing that for a smooth, invertible transformation $\mathbf{y} = \phi(\mathbf{x})$, the transformed score function $\nabla_{\mathbf{y}} \log q(\mathbf{y})$ can be expressed directly in terms of $\nabla_{\mathbf{x}} \log p(\mathbf{x})$. Using this result, we develop two applications: First, we establish a reverse-time It\^o lemma for score-based diffusion models, allowing the use of $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$ to reverse an SDE in the transformed space without directly learning $\nabla_{\mathbf{y}} \log q_t(\mathbf{y})$. This approach enables training diffusion models in one space but sampling in another, effectively decoupling the forward and reverse processes. Second, we introduce generalized sliced score matching, extending traditional sliced score matching from linear projections to arbitrary smooth transformations. This provides greater flexibility in high-dimensional density estimation. We demonstrate these theoretical advances through applications to diffusion on the probability simplex and empirically compare our generalized score matching approach against traditional sliced score matching methods.

Dense high dimensional vectors are becoming increasingly vital in fields such as computer vision, machine learning, and large language models (LLMs), serving as standard representations for multimodal data. Now the dimensionality of these vector can exceed several thousands easily. Despite the nearest neighbor search (NNS) over these dense high dimensional vectors have been widely used for retrieval augmented generation (RAG) and many other applications, the effectiveness of NNS in such a high-dimensional space remains uncertain, given the possible challenge caused by the "curse of dimensionality." To address above question, in this paper, we conduct extensive NNS studies with different distance functions, such as $L_1$ distance, $L_2$ distance and angular-distance, across diverse embedding datasets, of varied types, dimensionality and modality. Our aim is to investigate factors influencing the meaningfulness of NNS. Our experiments reveal that high-dimensional text embeddings exhibit increased resilience as dimensionality rises to higher levels when compared to random vectors. This resilience suggests that text embeddings are less affected to the "curse of dimensionality," resulting in more meaningful NNS outcomes for practical use. Additionally, the choice of distance function has minimal impact on the relevance of NNS. Our study shows the effectiveness of the embedding-based data representation method and can offer opportunity for further optimization of dense vector-related applications.

Causality inference is prone to spurious causal interactions, due to the substantial confounders in a complex system. While many existing methods based on the statistical methods or dynamical methods attempt to address misidentification challenges, there remains a notable lack of effective methods to infer causality, in particular in the presence of invisible/unobservable confounders. As a result, accurately inferring causation with invisible confounders remains a largely unexplored and outstanding issue in data science and AI fields. In this work, we propose a method to overcome such challenges to infer dynamical causality under invisible confounders (CIC method) and further reconstruct the invisible confounders from time-series data by developing an orthogonal decomposition theorem in a delay embedding space. The core of our CIC method lies in its ability to decompose the observed variables not in their original space but in their delay embedding space into the common and private subspaces respectively, thereby quantifying causality between those variables both theoretically and computationally. This theoretical foundation ensures the causal detection for any high-dimensional system even with only two observed variables under many invisible confounders, which is actually a long-standing problem in the field. In addition to the invisible confounder problem, such a decomposition actually makes the intertwined variables separable in the embedding space, thus also solving the non-separability problem of causal inference. Extensive validation of the CIC method is carried out using various real datasets, and the experimental results demonstrates its effectiveness to reconstruct real biological networks even with unobserved confounders.

In high-dimensional settings, Bayesian optimization (BO) can be expensive and infeasible. The random embedding Bayesian optimization algorithm is commonly used to address high-dimensional BO challenges. However, this method relies on the effective subspace assumption on the optimization problem's objective function, which limits its applicability. In this paper, we introduce Condensing-Expansion Projection Bayesian optimization (CEPBO), a novel random projection-based approach for high-dimensional BO that does not reply on the effective subspace assumption. The approach is both simple to implement and highly practical. We present two algorithms based on different random projection matrices: the Gaussian projection matrix and the hashing projection matrix. Experimental results demonstrate that both algorithms outperform existing random embedding-based algorithms in most cases, achieving superior performance on high-dimensional BO problems. The code is available in https://anonymous.4open.science/

Unsupervised embeddings are fundamental to numerous machine learning applications, yet their evaluation remains a challenging task. Traditional assessment methods often rely on extrinsic variables, such as performance in downstream tasks, which can introduce confounding factors and mask the true quality of embeddings. This paper introduces the Intrinsic Distance Preservation Evaluation (IDPE) method, a novel approach for assessing embedding quality based on the preservation of Mahalanobis distances between data points in the original and embedded spaces. We demonstrate the limitations of extrinsic evaluation methods through a simple example, highlighting how they can lead to misleading conclusions about embedding quality. IDPE addresses these issues by providing a task-independent measure of how well embeddings preserve the intrinsic structure of the original data. Our method leverages efficient similarity search techniques to make it applicable to large-scale datasets. We compare IDPE with established intrinsic metrics like trustworthiness and continuity, as well as extrinsic metrics such as Average Rank and Mean Reciprocal Rank. Our results show that IDPE offers a more comprehensive and reliable assessment of embedding quality across various scenarios. We evaluate PCA and t-SNE embeddings using IDPE, revealing insights into their performance that are not captured by traditional metrics. This work contributes to the field by providing a robust, efficient, and interpretable method for embedding evaluation. IDPE's focus on intrinsic properties offers a valuable tool for researchers and practitioners seeking to develop and assess high-quality embeddings for diverse machine learning applications.