DR may be supervised by taking advantage of class information.

Randomly pick α (0, 1) with probability density function f (α) = 2 α . Randomly pick α (0, 1) with probability density function f (α) = 2 α . We dedicate the rest of the section to prove Theorem 4. Notice that Algorithm 4 is identical to Theorem 4 follows by the exact same steps that Theorem 3 follows using Lemma 2. The proof of Lemma 4 is concluded at the end of the section. Lemma 5. [36] F or the matrix B constructed at Step 2 of Algorithm 4, the following holds: 1. This is formally stated below.

The widely studied Generalized Min-Sum-Set-Cover (GMSSC) problem serves as a formal model for the setting above.

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.

Dimensionality reduction (DR) and deep neural networks (DNNs) are two important aspects in data analysis. In data analysis and deep learning, the datasets are often high-dimensional and exhibit some complicated topological structures due to various backgrounds from science to engineering [1,2,4-7]. Traditional approaches to data analysis and visualization, in particular on images recognition, often fail in the high-dimensional setting, and a common practice is to perform dimensionality reduction [2, 6, 11] in order to make data analysis tractable and economic, and the DNNs is a powerful tool in dealing with non-linear dimensionality reduction problems. It has now been recognized that practical datasets often consists of features of low intrinsic dimensions with some nontrivial topological structures [1,2,6], and the geometric structure of datasets heavily affect the architecture of the deep neural networks. Nonetheless, how and to what extent the geometric (topological) structure of datasets is connected with the architecture of a deep neural network remains unclear and is an active research area of deep learning in recent years. 1

Abstract--Data dimensionality reduction techniques are often utilized in the implementation of Quantum Machine Learning models to address two significant issues: the constraints of NISQ quantum devices, which are characterized by noise and a limited number of qubits, and the challenge of simulating a large number of qubits on classical devices. It also raises concerns over the scalability of these approaches, as dimensionality reduction methods are slow to adapt to large datasets. In this article, we analyze how data reduction methods affect different QML models. We conduct this experiment over several generated datasets, quantum machine algorithms, quantum data encoding methods, and data reduction methods. All these models were evaluated on the performance metrics like accuracy, precision, recall, and F1 score. Our findings have led us to conclude that the usage of data dimensionality reduction methods results in skewed performance metric values, which results in wrongly estimating the actual performance of quantum machine learning models. There are several factors, along with data dimensionality reduction methods, that worsen this problem, such as characteristics of the datasets, classical to quantum information embedding methods, percentage of feature reduction, classical components associated with quantum models, and structure of quantum machine learning models. We consistently observed the difference in the accuracy range of 14% to 48% amongst these models, using data reduction and not using it. Apart from this, our observations have shown that some data reduction methods tend to perform better for some specific data embedding methodologies and ansatz constructions. In recent decades, there has been a significant push towards research and development of Quantum Machine Learning algorithms and models. Quantum Machine Learning has also been heralded as one of the prominent use cases for Quantum Computing devices. Several studies have shown the ability of QML models to solve difficult machine-learning problems and sometimes outperform the classical approach. Mostly, these proofs are either theoretical or simulated on classical devices. This is because the current quantum computational devices lack the required number of qubits, have questionable error correction ability, and tend to have noisy qubits.

The Internet of Things (IoT) plays a major role today in smart building infrastructures, from simple smart-home applications, to more sophisticated industrial type installations. The vast amounts of data generated from relevant systems can be processed in different ways revealing important information. This is especially true in the era of edge computing, when advanced data analysis and decision-making is gradually moving to the edge of the network where devices are generally characterised by low computing resources. In this context, one of the emerging main challenges is related to maintaining data analysis accuracy even with less data that can be efficiently handled by low resource devices. The present work focuses on correlation analysis of data retrieved from a pilot IoT network installation monitoring a small smart office by means of environmental and energy consumption sensors. The research motivation was to find statistical correlation between the monitoring variables that will allow the use of machine learning (ML) prediction algorithms for energy consumption reducing input parameters. For this to happen, a series of hypothesis tests for the correlation of three different environmental variables with the energy consumption were carried out. A total of ninety tests were performed, thirty for each pair of variables. In these tests, p-values showed the existence of strong or semi-strong correlation with two environmental variables, and of a weak correlation with a third one. Using the proposed methodology, we manage without examining the entire data set to exclude weak correlated variables while keeping the same score of accuracy.

Bayesian optimisation (BO) is a standard approach for sample-efficient global optimisation of expensive black-box functions, yet its scalability to high dimensions remains challenging. Here, we investigate nonlinear dimensionality reduction techniques that reduce the problem to a sequence of low-dimensional Latent-Space BO (LSBO). While early LSBO methods used (linear) random projections (Wang et al., 2013), building on Grosnit et al. (2021), we employ Variational Autoencoders (VAEs) for LSBO, focusing on deep metric loss for structured latent manifolds and VAE retraining to adapt the encoder-decoder to newly sampled regions. We propose some changes in their implementation, originally designed for tasks such as molecule generation, and reformulate the algorithm for broader optimisation purposes. We then couple LSBO with Sequential Domain Reduction (SDR) directly in the latent space (SDR-LSBO), yielding an algorithm that narrows the latent search domains as evidence accumulates. Implemented in a GPU-accelerated BoTorch stack with Matern-5/2 Gaussian process surrogates, our numerical results show improved optimisation quality across benchmark tasks and that structured latent manifolds improve BO performance. Additionally, we compare random embeddings and VAEs as two mechanisms for dimensionality reduction, showing that the latter outperforms the former. To the best of our knowledge, this is the first study to combine SDR with VAE-based LSBO, and our analysis clarifies design choices for metric shaping and retraining that are critical for scalable latent space BO. For reproducibility, our source code is available at https://github.com/L-Lok/Nonlinear-Dimensionality-Reduction-Techniques-for-Bayesian-Optimization.git.

Despite their growing popularity, there remains a prevalent misconception among practitioners about the equivalence in performance between parametric and non-parametric methods.

Despite its broad applications in fields such as computer vision, graph learning, and natural language processing, the development of a data projection model that can be effectively used to visualize data in the context of FL is crucial yet remains heavily under-explored. Neighbor embedding (NE) is an essential technique for visualizing complex high-dimensional data, but collab-oratively learning a joint NE model is difficult.