Principal Subspaces Research Articles

Jet schemes, quantum dilogarithm and Feigin-Stoyanovsky's principal subspaces

We analyze the structure of Feigin-Stoyanovsky's principal subspaces of affine Lie algebra from the jet algebra viewpoint. For type A level one principal subspaces, we show that their shifted multi-graded Hilbert series can be expressed either using the quantum dilogarithm or as certain generating functions “counting” finite-dimensional representations of A-type quivers. This notably results in novel fermionic character formulas for these principal subspaces. Moreover, our result implies that all level one principal subspaces of type A are “classically free” as vertex algebras.We also analyze infinite jet algebras associated to principal subspaces of affine vertex algebras L1(so5), L1(so8) and L1(g2). We derive a new character formula for the principal subspace of L1(so5), proving that it is classically free, and present evidence that the principal subspaces of L1(so8) and of L1(g2) are also classically free.

Combinatorial bases of standard modules of twisted affine Lie algebras in types and : rectangular highest weights

We consider the standard modules of rectangular highest weights of affine Lie algebras in types and . By using vertex algebraic techniques we construct the combinatorial bases for standard modules and their principal subspaces and parafermionic spaces. Finally, we compute the corresponding character formulae and, as an application, we obtain two new families of combinatorial identities.

CX-DaGAN: Domain Adaptation for Pneumonia Diagnosis on a Small Chest X-Ray Dataset.

Recent advances in deep learning led to several algorithms for the accurate diagnosis of pneumonia from chest X-rays. However, these models require large training medical datasets, which are sparse, isolated, and generally private. Furthermore, these models in medical imaging are known to over-fit to a particular data domain source, i.e., these algorithms do not conserve the same accuracy when tested on a dataset from another medical center, mainly due to image distribution discrepancies. In this work, a domain adaptation and classification technique is proposed to overcome the over-fit challenges on a small dataset. This method uses a private-small dataset (target domain), a public-large labeled dataset from another medical center (source domain), and consists of three steps. First, it performs a data selection of the source domain's most representative images based on similarity constraints through principal component analysis subspaces. Second, the selected samples from the source domain are fit to the target distribution through an image to image translation based on a cycle-generative adversarial network. Finally, the target train dataset and the adapted images from the source dataset are used within a convolutional neural network to explore different settings to adjust the layers and perform the classification of the target test dataset. It is shown that fine-tuning a few specific layers together with the selected-adapted images increases the sorting accuracy while reducing the trainable parameters. The proposed approach achieved a notable increase in the target dataset's overall classification accuracy, reaching up to 97.78 % compared to 90.03 % by standard transfer learning.

Parafermionic Bases of Standard Modules for Twisted Affine Lie Algebras of Type $A_{2l-1}^{(2)}$, $D_{l+1}^{(2)}$, $E_{6}^{(2)}$ and $D_{4}^{(3)}$

Using the bases of principal subspaces for twisted affine Lie algebras except \(A_{2l}^{(2)}\) by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight kΛ0 and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in Hatayama et al. (2001).

Principal subspaces for the affine Lie algebras in types D, E and F

We consider the principal subspaces of certain level $k\geqslant 1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.

Principal subspaces for the quantum affine vertex algebra in type A1(1)

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers-Ramanujan-type identities.

CTRL: Closed-Loop Transcription to an LDR via Minimaxing Rate Reduction.

This work proposes a new computational framework for learning a structured generative model for real-world datasets. In particular, we propose to learn a Closed-loop Transcriptionbetween a multi-class, multi-dimensional data distribution and a Linear discriminative representation (CTRL) in the feature space that consists of multiple independent multi-dimensional linear subspaces. In particular, we argue that the optimal encoding and decoding mappings sought can be formulated as a two-player minimax game between the encoder and decoderfor the learned representation. A natural utility function for this game is the so-called rate reduction, a simple information-theoretic measure for distances between mixtures of subspace-like Gaussians in the feature space. Our formulation draws inspiration from closed-loop error feedback from control systems and avoids expensive evaluating and minimizing of approximated distances between arbitrary distributions in either the data space or the feature space. To a large extent, this new formulation unifies the concepts and benefits of Auto-Encoding and GAN and naturally extends them to the settings of learning a both discriminative and generative representation for multi-class and multi-dimensional real-world data. Our extensive experiments on many benchmark imagery datasets demonstrate tremendous potential of this new closed-loop formulation: under fair comparison, visual quality of the learned decoder and classification performance of the encoder is competitive and arguably better than existing methods based on GAN, VAE, or a combination of both. Unlike existing generative models, the so-learned features of the multiple classes are structured instead of hidden: different classes are explicitly mapped onto corresponding independent principal subspaces in the feature space, and diverse visual attributes within each class are modeled by the independent principal components within each subspace.

Exploring the common principal subspace of deep features in neural networks

We find that different Deep Neural Networks (DNNs) trained with the same dataset share a common principal subspace in latent spaces, no matter in which architectures (e.g., Convolutional Neural Networks (CNNs), Multi-Layer Preceptors (MLPs) and Autoencoders (AEs)) the DNNs were built or even whether labels have been used in training (e.g., supervised, unsupervised, and self-supervised learning). Specifically, we design a new metric $${\mathcal {P}}$$ -vector to represent the principal subspace of deep features learned in a DNN, and propose to measure angles between the principal subspaces using $${\mathcal {P}}$$ -vectors. Small angles (with cosine close to 1.0) have been found in the comparisons between any two DNNs trained with different algorithms/architectures. Furthermore, during the training procedure from random scratch, the angle decrease from a larger one (70°–80° usually) to the small one, which coincides the progress of feature space learning from scratch to convergence. Then, we carry out case studies to measure the angle between the $${\mathcal {P}}$$ -vector and the principal subspace of training dataset, and connect such angle with generalization performance. Extensive experiments with practically-used Multi-Layer Perceptron (MLPs), AEs and CNNs for classification, image reconstruction, and self-supervised learning tasks on MNIST, CIFAR-10 and CIFAR-100 datasets have been done to support our claims with solid evidences.

Updating the Multilinear UTV Decomposition

Low multilinear rank approximation (LMLRA) is a basic tool for compressing a tensor into a more compact form, while preserving most of its information. An LMLRA of an <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>th order tensor consists of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> factor matrices which are combined using a core tensor of smaller dimensions than the original tensor. When the tensor is non-static, for example when over time new tensor slices are being added along a certain mode, one can expect that the LMLRA of the tensor changes, but that an approximation of the new tensor can be derived from the previous one. By updating the factor matrices and the core tensor in an efficient way when a new tensor slice is added, this can indeed be achieved. In this paper, a new method is proposed to track the factor matrices in both the evolving and non-evolving modes, leading to efficient and accurate updates of the LMLRA. We track a truncated multilinear UTV decomposition (TMLUTVD), which is a tensor decomposition that is multilinear rank-revealing, yet less expensive to compute than the popular multilinear singular value decomposition. We derive accuracy bounds for the tracked TMLUTVD and give strategies to speed up its computation. Experiments show that the TMLUTVD tracking method works well in practice, even if the mode-<inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> principal subspaces change drastically between updates. As an illustration of the possibilities of the technique, we extend the use of LMLRA in a compression step, prior to the computation of a canonical polyadic decomposition (CPD), to the setting of updating and tracking.

Some remarks on associated varieties of vertex operator superalgebras

We study several families of vertex operator superalgebras from a jet (super)scheme point of view. We provide new examples of vertex algebras which are “chiral-quantizations" of their \(C_{2}\)-algebras \(R_V\). Our examples come from affine \(C_\ell ^{(1)}\)-series vertex algebras, \(\ell \geqslant 1\), certain \(N=1\) superconformal vertex algebras, Feigin–Stoyanovsky principal subspaces, Feigin–Stoyanovsky type subspaces, graph vertex algebras \(W_{\Gamma }\), and extended Virasoro vertex algebras. We also give a counterexample to the chiral-quantization property for the \(N=2\) superconformal vertex algebra with central charge 1.

The projective geometry over partially ordered skew fields, II

In this paper > the investigation of properties for partially ordered linear spaces over partially ordered skew fields is prolonged. This investigation was started in part I >. Derivative lattices associated partially ordered linear spaces over partially ordered skew fields are examined. More exactly, properties of the convex projective geometry $${\cal L}$$ of a partially ordered linear space $${}_FV$$ over a partially ordered skew field $$F$$ are considered. The convexity of linear subspaces has meaning the Abelian convexity ($$ab$$-convexity), which is based on the definition of a convex subgroup for a partially ordered group. Second and third theorems of linear spaces order isomorphisms for interpolation linear spaces over partially ordered skew fields are proved. Some theorems are proved for principal linear subspaces of interpolation linear spaces over directed skew fields. The principal linear subspace $$I_a$$ of a partially ordered linear space $${}_FV$$ over a partially ordered skew field $$F$$ is the smallest $$ab$$-convex directed linear subspace of linear space $${}_FV$$ which contains the positive element $$a\in V$$. The analog for the third theorem of linear spaces order isomorphisms for principal linear subspaces is demonstrated in interpolation linear spaces over directed skew fields

Parafermionic bases of standard modules for affine Lie algebras

In this paper we construct combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the untwisted affine Lie algebras. Our construction is a modification of G. Georgiev's construction for the affine Lie algebra $\widehat{\mathfrak sl}(n+1,\mathbb C)$---the constructed parafermionic bases are projections of the quasi-particle bases of the principal subspaces, obtained previously in a series of papers by the first two authors. As a consequence we prove the character formula of A. Kuniba, T. Nakanishi and J. Suzuki for all non-simply-laced untwisted affine Lie algebras.

Subspace Energy Monitoring for Anomaly Detection @Sensor or @Edge

The amount of data generated by distributed monitoring systems that can be exploited for anomaly detection, along with real time, bandwidth, and scalability requirements leads to the abandonment of centralized approaches in favor of processing closer to where data are generated. This increases the interest in algorithms coping with the limited computational resources of gateways or sensor nodes. We here propose two dual and lightweight methods for anomaly detection based on generalized spectral analysis. We monitor the signal energy laying along with the principal and anti -principal signal subspaces, and call for an anomaly when such energy changes significantly with respect to normal conditions. A streaming approach for the online estimation of the needed subspaces is also proposed. The methods are tested by applying them to synthetic data and real-world sensor readings. The synthetic setting is used for design space exploration and highlights the tradeoff between accuracy and computational cost. The real-world example deals with structural health monitoring and shows how, despite the extremely low computations costs, our methods are able to detect permanent and transient anomalies that would classically be detected by full spectral analysis.

A note on principal subspaces of the affine Lie algebras in types and

We construct quasi-particle bases of principal subspaces of standard modules where and Λj denotes the fundamental weight of affine Lie algebras of type or of level one. From the given bases we find characters of principal subspaces.

Projection pursuit based tests of normality with functional data

Methods for validating the assumption of normality of functional data have been only lightly developed to date, with existing methods based primarily on summarizing the data by their projections into random or principal component subspaces, and applying multivariate normality tests to the vectors of scores defining these projections. While this is effective in some cases, we show with both real and synthetic data examples some pitfalls of this approach, including their sensitivity to the basis used to smooth the raw data. We propose a new normality test for functional data based on a projection pursuit that overcomes some of these challenges. Asymptotic theory is developed for the proposed statistics, and we develop several new computational tools needed to implement the high-dimensional projection pursuit. As a by-product of evaluating the test statistic, our method furnishes a way of decomposing functional data into its approximately Gaussian and non-Gaussian components, which is useful for the purpose of data visualization and subsequent analyses. A simulation study and analysis of three data sets demonstrate the complimentary advantages of the proposed test to those currently available in the literature.

Spaces invariant under unitary representations of discrete groups

We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an isometry intertwining the representation with the right regular representation, that we call a Helson map. We then characterize invariant subspaces using a Helson map, and provide general characterizations of Riesz and frame sequences of orbits. These results extend to the nonabelian setting several known results for abelian groups. They also extend to countable families of generators previous results obtained for principal subspaces.

Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning.

Principal component analysis (PCA) and Kernel principal component analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both (finite and infinite) situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by K-tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets.

Semi-Supervised Robust Mixture Models in RKHS for Abnormality Detection in Medical Images

Abnormality detection in medical images is a one-class classification problem for which existing methods typically involve variants of kernel principal component analysis or one-class support vector machines. However, existing methods rely on highly-curated training sets with full supervision, often using heuristics for model fitting or ignore the variances of the data within principal subspaces. In contrast, we propose novel methods that can work with imperfectly curated datasets using robust statistical learning, by extending the multivariate generalized-Gaussian distribution to a reproducing kernel Hilbert space (RKHS) and employing it within a mixture model. We propose a novel semi-supervised extension of our learning scheme, showing that a small amount of expert feedback through high-quality labeled data of the outlier class can boost performance. We propose expectation maximization for our semi-supervised robust mixture-model learning in RKHS, using solely the Gram matrix and without the explicit lifting map. Our methods incorporate optimal component means, principal directions, and variances for abnormality detection. Results on four large public datasets on retinopathy and cancer, compared against a variety of contemporary methods, show that our method gives benefits over the state of the art in one-class classification for abnormality detection.

Online monitoring scheme using principal component analysis through Kullback-Leibler divergence analysis technique for fault detection

Principal component analysis (PCA) is a common tool in the literature and widely used for process monitoring and fault detection. Traditional PCA is associated with the two well-known control charts, the Hotelling’s T2 and the squared prediction error (SPE), as monitoring statistics. This paper develops the use of new measures based on a distribution dissimilarity technique named Kullback-Leibler divergence (KLD) through PCA by measuring the difference between online estimated and offline reference density functions. For processes with PCA scores following a multivariate Gaussian distribution, KLD is computed on both principal and residual subspaces defined by PCA in a moving window to extract the local disparity information. The potentials of the proposed algorithm are afterwards demonstrated through an application on two well-known processes in chemical industries; the Tennessee Eastman process as a reference benchmark and three tank system as an experimental validation. The monitoring performance was compared to recent results from other multivariate statistical process monitoring (MSPM) techniques. The proposed method showed superior robustness and effectiveness recording the lowest average missed detection rate and false alarm rates in process fault detection.

Prototype learning and collaborative representation using Grassmann manifolds for image set classification

Image set classification using manifolds is becoming increasingly more attractive since it considers non-Euclidean geometry. However, with the success of dictionary learning for image set classification using manifolds, how to learn an over-complete dictionary is still challenging. This paper proposes a novel prototype subspace learning method, in which a set of images is represented by a linear subspace and then mapped onto a Grassmann manifold. With this subspace representation, class prototypes and intra-class differences can be represented as principal components and variation subspaces, respectively. Isometric mapping further maps the manifolds into the symmetrical space via collaborative representation, which permits a closed-term solution. The proposed method is evaluated for face recognition, object recognition and action recognition. Extensive experimental results on the Honda, Extended YaleB, ETH-80 and Cambridge-Gesture datasets verify the superiority of the proposed method over the state-of-the-art methods.