相关论文: Measuring the Data

In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results…

Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display…

One of the main postulates of quantum mechanics is that measurements destroy quantum coherence (wave function collapse). Recently it was discovered that in a many-body system dilute local measurements still preserve some coherence across…

The ongoing exponential rise in recording capacity calls for new approaches for analysing and interpreting neural data. Effective dimensionality has emerged as an important property of neural activity across populations of neurons, yet…

We propose a novel algorithm for supervised dimensionality reduction named Manifold Partition Discriminant Analysis (MPDA). It aims to find a linear embedding space where the within-class similarity is achieved along the direction that is…

Diffusion models are powerful deep generative models, but unlike classical models, they lack an explicit low-dimensional latent space that parameterizes the data manifold. This absence makes it difficult to perform manifold-aware…

The manifold of empirical mean values of statistical data ad infinitum has a geometric shape that depends on the probability measure that governs the generating model. Large deviation theory produces entropy functions that depend on both…

An important theme in modern inverse problems is the reconstruction of time-dependent data from only finitely many measurements. To obtain satisfactory reconstruction results in this setting it is essential to strongly exploit temporal…

For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS)…

Observational entropy is interpreted as the uncertainty an observer making measurements associates with a system. So far, properties that make such an interpretation possible rely on the assumption of ideal projective measurements. We show…

High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of…

The problems of computational data processing involving regression, interpolation, reconstruction and imputation for multidimensional big datasets are becoming more important these days, because of the availability of data and their widely…

It is widely believed that natural image data exhibits low-dimensional structure despite the high dimensionality of conventional pixel representations. This idea underlies a common intuition for the remarkable success of deep learning in…

Motivated by the problem of selling large, proprietary data, we consider an information pricing problem proposed by Bergemann et al. that involves a decision-making buyer and a monopolistic seller. The seller has access to the underlying…

Manifold learning techniques have become increasingly valuable as data continues to grow in size. By discovering a lower-dimensional representation (embedding) of the structure of a dataset, manifold learning algorithms can substantially…

Today, with the growing demands of information storage and data transfer, data compression is becoming increasingly important. Data Compression is a technique which is used to decrease the size of data. This is very useful when some huge…

In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets.…

Transformers are deep neural network architectures that underpin the recent successes of large language models. Unlike more classical architectures that can be viewed as point-to-point maps, a Transformer acts as a measure-to-measure map…

The curse of dimensionality is a phenomenon frequently observed in machine learning (ML) and knowledge discovery (KD). There is a large body of literature investigating its origin and impact, using methods from mathematics as well as from…

Information about intrinsic dimension is crucial to perform dimensionality reduction, compress information, design efficient algorithms, and do statistical adaptation. In this paper we propose an estimator for the intrinsic dimension of a…