Dimensional Subspace Research Articles

Environmental sound classification using raw-audio based ensemble framework

Environmental sound classification (ESC) aims to automatically recognize audio recordings from the underlying environment, such as "urban park" or "city centre". Most of the existing methods for ESC use hand-crafted time-frequency features such as log-mel spectrogram to represent audio recordings. However, the hand-crafted features rely on transformations that are defined beforehand and do not consider the variability in the environment due to differences in recording conditions or recording devices. To overcome this, we present an alternative representation framework by leveraging a pre-trained convolutional neural network, SoundNet, trained on a large-scale audio dataset to represent raw audio recordings. We observe that the representations obtained from the intermediate layers of SoundNet lie in low-dimensional subspace. However, the dimensionality of the low-dimensional subspace is not known. To address this, an automatic compact dictionary learning framework is utilized that gives the dimensionality of the underlying subspace. The low-dimensional embeddings are then aggregated in a late-fusion manner in the ensemble framework to incorporate hierarchical information learned at various intermediate layers of SoundNet. We perform experimental evaluation on publicly available DCASE 2017 and 2018 ASC datasets. The proposed ensemble framework improves performance between 1 and 4 percentage points compared to that of existing time-frequency representations.

An auxiliary correction model implemented by the Correction Property of intermediate layer against adversarial examples

In powerful adversarial attacks against deep neural networks (DNN), the generated adversarial example will mislead the DNN-implemented classifier by destroying the features of the last layer. To enhance the robustness of the classifier, in our paper, a Feature Analysis and Conditional Matching prediction distribution (FACM) model is proposed to utilize the features of intermediate layers to correct the misclassification. Specifically, we first prove that the intermediate layers of the classifier still retain effective features for the original category when the classifier is subjected to adversarial attacks, which is defined as the Correction Property in our paper. According to this, we propose the FACM model consisting of Feature Analysis (FA) correction module, Conditional Matching Prediction Distribution (CMPD) correction module and decision module. Specifically, the FA correction module is comprised of fully connected layers, which takes the features of the intermediate layers as the input to correct the misclassification of the classifier. The CMPD correction module is based on a conditional autoencoder, which not only uses the features of intermediate layers as the condition to accelerate convergence but also mitigates the negative effect of adversarial examples, trained with the Kullback–Leibler loss to match prediction distribution. Through the empirically verified Diversity Property among the individual correction modules, the decision module is proposed to integrate the proposed correction modules to enhance the DNN-implemented classifier’s robustness by reducing the dimensionality of adversarial subspace. That is, the input perturbed in certain directions (i.e., dimensions) that lead to misclassifications for the classifier can be correctly classified by the proposed correction modules. The extended experiments demonstrate our FACM model outperforms the existing methods against adversarial attacks, especially optimization-based white-box attacks and query-based black-box attacks.

Light field images super-resolution method based on hybrid low-dimensional spatial–angular interaction feature and linear complementation epipolar feature

Light-field (LF) cameras record 4D information about the intensity and angle of light, but limited by the resolution of the sensor, LF images require a trade-off between spatial and angular resolution, which results in generally low spatial resolution for LF images. We work on spatial super-resolution reconstruction (SR) of LF images, but due to the high dimensionality of 4D LF, it is difficult to process it directly, so we decompose 4D LF into low-dimensional sub-spaces. Since the 4D features of interest to different low-dimensional sub-spaces are different, we design a hybrid network structure—HSAESR, which processes multiple low-dimensional sub-spaces individually to adequately model the texture information in the 4D LF. HSAESR is designed to reconstruct Macro Pixel Images (MacPI) and Extremely Planar Images (EPI) respectively, and the final reconstruction results are weighted and fused. For MacPI, a CNN-based feature interaction module is designed using a novel asymptotic ’grid’ interaction approach to fully utilize the correlation between the 2D spatial dimension and the 2D angular dimension sub-spaces in 4D LF. For EPI, a Transformer-based epipolar feature linear complementation module is designed learning global and non-global multi-directional epipolar feature correlations, which learns feature linear correlation factor (α) and linear bias weight (β) through the correlation of non-global epipolar features, and corrects and complements the global epipolar correlation features through feature linear complementation. Extensive experiments are conducted on five datasets, the results show that HSAESR further improves the performance of LF images spatial SR, the reconstruction results are better than the comparison methods.

Understanding overfitting in random forest for probability estimation: a visualization and simulation study

BackgroundRandom forests have become popular for clinical risk prediction modeling. In a case study on predicting ovarian malignancy, we observed training AUCs close to 1. Although this suggests overfitting, performance was competitive on test data. We aimed to understand the behavior of random forests for probability estimation by (1) visualizing data space in three real-world case studies and (2) a simulation study.MethodsFor the case studies, multinomial risk estimates were visualized using heatmaps in a 2-dimensional subspace. The simulation study included 48 logistic data-generating mechanisms (DGM), varying the predictor distribution, the number of predictors, the correlation between predictors, the true AUC, and the strength of true predictors. For each DGM, 1000 training datasets of size 200 or 4000 with binary outcomes were simulated, and random forest models were trained with minimum node size 2 or 20 using the ranger R package, resulting in 192 scenarios in total. Model performance was evaluated on large test datasets (N = 100,000).ResultsThe visualizations suggested that the model learned “spikes of probability” around events in the training set. A cluster of events created a bigger peak or plateau (signal), isolated events local peaks (noise). In the simulation study, median training AUCs were between 0.97 and 1 unless there were 4 binary predictors or 16 binary predictors with a minimum node size of 20. The median discrimination loss, i.e., the difference between the median test AUC and the true AUC, was 0.025 (range 0.00 to 0.13). Median training AUCs had Spearman correlations of around 0.70 with discrimination loss. Median test AUCs were higher with higher events per variable, higher minimum node size, and binary predictors. Median training calibration slopes were always above 1 and were not correlated with median test slopes across scenarios (Spearman correlation − 0.11). Median test slopes were higher with higher true AUC, higher minimum node size, and higher sample size.ConclusionsRandom forests learn local probability peaks that often yield near perfect training AUCs without strongly affecting AUCs on test data. When the aim is probability estimation, the simulation results go against the common recommendation to use fully grown trees in random forest models.

The maximum principle for lumped-distributed control systems.

This paper concerns the optimal control of lumped-distributed systems, an examplar of which is a mechanical system with rotating components connected by flexible rods. The underlying mathematical model is a controlled semilinear evolution equation, in which nonlinear terms involve a projection of the full state onto a finite dimensional subspace. We derive necessary conditions of optimality in the form of a maximum principle, for a problem formulation which involves pathwise and end-point constraints on the lumped components of the state variable. Perturbational methods are employed in the derivation, which allow for non-smooth data. A key step in the analysis is the reduction of the optimization problem with an infinite dimensional state space, to one with finite dimensional aspects. The computational implications of this reduction will be explored in future work. Necessary conditions for problems with state constraints and end-point constraints can only be achieved under rather strict compatibility hypotheses on the way the dynamics interact with the state constraint and end-point constraint sets, due the infinite dimensionality of the state space. This paper identifies a class of infinite dimensional problems, of engineering significance, for which necessary conditions of optimality can be derived, in the absence of any additional compatibility hypotheses.

Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns

We propose a general method for constructing a minimal cover of high-dimensional chaotic attractors by embedded unstable recurrent patterns. By minimal cover we mean a subset of available patterns such that the approximation of chaotic dynamics by a minimal cover with a predefined proximity threshold is as good as the approximation by the full available set. The proximity measure, based on the concept of a directed Hausdorff distance, can be chosen with considerable freedom and adapted to the properties of a given chaotic system. In the context of a spatiotemporally chaotic attractor of the Kuramoto–Sivashinsky system on a periodic domain, we demonstrate that the minimal cover can be faithfully constructed even when the proximity measure is defined within a subspace of dimension much smaller than the dimension of space containing the attractor. We discuss how the minimal cover can be used to provide a reduced description of the attractor structure and the dynamics on it.

Stereologically adapted Crofton formulae for tensor valuations

The classical Crofton formula explains how intrinsic volumes of a convex body K in n-dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with invariantly moved affine flats. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.

Null states and time evolution in a toy model of black hole dynamics

Spacetime wormholes can provide non-perturbative contributions to the gravitational path integral that make the actual number of states eS in a gravitational system much smaller than the number of states eSp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{S_{\ extrm{p}}} $$\\end{document} predicted by perturbative semiclassical effective field theory. The effects on the physics of the system are naturally profound in contexts in which the perturbative description actively involves N = O(eS) of the possible eSp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{S_{\ extrm{p}}} $$\\end{document} perturbative states; e.g., in late stages of black hole evaporation. Such contexts are typically associated with the existence of non-trivial quantum extremal surfaces. However, by forcing a simple topological gravity model to evolve in time, we find that such effects can also have large impact for N ≪ eS (in which case no quantum extremal surfaces can arise). In particular, even for small N, the insertion of generic operators into the path integral can cause the non-perturbative time evolution to differ dramatically from perturbative expectations. On the other hand, this discrepancy is small for the special case where the inserted operators are non-trivial only in a subspace of dimension D ≪ eS. We thus study this latter case in detail. We also discuss potential implications for more realistic gravitational systems.

A type of multigrid method for semilinear elliptic problems based on symmetric interior penalty discontinuous Galerkin method

AbstractThis article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.

Some Properties of Subspaces over Residue Class Rings

Some Properties of Subspaces over Residue Class Rings

On one-sided testing affine subspaces

We study the query complexity of one-sided ϵ-testing the class of Boolean functions f:Fn→{0,1} that describe affine subspaces and Boolean functions that describe axis-parallel affine subspaces, where F is any finite field. We give polynomial-time ϵ-testers that ask O˜(1/ϵ) queries. This improves the query complexity O˜(|F|/ϵ) in [14]. The almost optimality of the algorithms follows from the lower bound of Ω(1/ϵ) for the query complexity proved by Bshouty and Goldreich [3].We then show that any one-sided ϵ-tester with proximity parameter ϵ<1/|F|d for the class of Boolean functions that describe (n−d)-dimensional affine subspaces and Boolean functions that describe axis-parallel (n−d)-dimensional affine subspaces must make at least Ω(1/ϵ+|F|d−1log⁡n) and Ω(1/ϵ+|F|d−1n) queries, respectively. This improves the lower bound Ω(log⁡n/log⁡log⁡n) that is proved in [14] for F=GF(2). We also give testers for those classes with query complexity that almost match the lower bounds.1

Two-dimensional and absolutely entanglement-breaking subspaces

Entanglement-breaking (EB) subspaces determine the additivity of entanglement of formation (EOF), which is a long-standing issue in quantum information. We explicitly construct the two-dimensional EB subspaces of any bipartite system, when system dimensions are equal, and we apply the subspaces to construct EB spaces of arbitrary dimensions. We also present partial construction when system dimensions are different. Then, we present the notion and properties of EB subspaces for some systems, and in particular the absolute EB subspaces. We construct some examples of absolute EB subspaces, as well as EB subspaces for some systems by using multiqubit Dicke states.

The controllability and structural controllability of Laplacian dynamics

In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix L ∗ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs; however, it is always simple for the Laplacian L with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with L. For a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with L ∗ . In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between parent nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from parent nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant. Finally, we present a sufficient and necessary condition to identify the strong structural controllability. Moreover, a method to construct a strongly structurally controllable graph is proposed.

Generalised evasive subspaces

AbstractWe introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a nonconstructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank‐metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank‐metric codes.

Projected data assimilation using sliding window proper orthogonal decomposition

Prediction of the state evolution of complex high-dimensional nonlinear systems is challenging due to the nonlinear sensitivity of the evolution to small inaccuracies in the model. Data Assimilation (DA) techniques improve state estimates by combining model simulations with real-time data. Few DA techniques can simultaneously handle nonlinear evolution, non-Gaussian uncertainty, and the high dimension of the state. We recently proposed addressing these challenges using a POD technique that projects the physical and data models into a reduced-dimensional subspace. Proper Orthogonal Decomposition (POD) is a tool to extract spatiotemporal patterns (modes) that dominate the observed data. We combined the POD-based projection operator, computed in an offline fashion, with a DA scheme that models non-Gaussian uncertainty in lower dimensional subspace. If the model parameters change significantly during time evolution, the offline computation of the projection operators ceases to be useful. We address this challenge using a Sliding Window POD (SW-POD), which recomputes the projection operator based on a sliding subset of snapshots from the entire evolution. The physical model projection is updated dynamically in terms of modes and number of modes, and the data model projection is also chosen to promote a sparse approximation. We test the efficacy of this technique on a modified Lorenz'96 model (L96) with a time-varying forcing and compare it with the time-invariant offline projected algorithm. In particular, dynamically determined physical and data model projections decrease the Root Mean Squared Error (RMSE) and the resampling rate.

Cyclic 2-spreads in V(6,q) and flag-transitive linear spaces

In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results.

Dual contrastive learning for multi-view clustering

Self-representation subspace clustering, based on the self-representation property of linear subspaces, has demonstrated superior performance in subspace clustering. However, many existing methods learn self-representation in the embedding space, which fails to accurately capture the clustering structure of the data. Additionally, in the data embedding process, only the shared features among clusters are removed, and the discriminative features that distinguish different clusters cannot be effectively selected. To address these limitations, we propose a contrastive learning-based self-representation subspace clustering method named Multi-view Contrastive Subspace Clustering (MCSC). This method integrates feature selection and self-representation learning into a unified framework. It learns a shared self-representation coefficient in both the embedding space and the original space using contrastive learning, which allows for a more accurate description of the linear relationship between samples and a better depiction of the inter-sample structure simultaneously. Moreover, we employ the ℓ1,2-norm for feature selection, which eliminates shared redundancies while preserving important features specific to each cluster, making the samples more distinguishable. Extensive experiments on six datasets verify the validity of our proposed model.

Geometry of entanglement and separability in Hilbert subspaces of dimension up to three

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. https://doi.org/10.1103/PhysRevA.95.032308), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.

Alternating multiple mixed values: Regularization, special values, parity, and dimension conjectures

In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple M-values (AMMVs), forming a Q[i]-subspace of the colored MZVs of level four. This variant includes the alternating version of Hoffman’s multiple t-values, Kaneko–Tsumura’s multiple T-values, and the multiple S-values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the generalized duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs by adopting the machinery from color MZVs of level four. As an important application, we prove a parity result for AMMVs previously conjectured by two of us. We also investigate several alternating multiple S- and T-values by establishing some explicit relations of integrals involving arctangent function. At the end, we compute the dimensions of a few interesting subspaces of AMMVs for weight less than 9. Supported by theoretical and numerical evidence aided by numerical and symbolic computation, we formulate a few conjectures concerning the dimensions of the above-mentioned subspaces of AMMVs. These conjectures hint at a few very rich but previously overlooked algebraic and geometric structures associated with these vector spaces.

On controllability of temporal networks

Temporality has been recently identified as a useful feature to exploit when controlling a complex network. Empirical evidence has in fact shown that, with respect to their static counterparts, temporal networks (i) are often endowed with larger reachable sets and (ii) require less control energy when steered towards an arbitrary target state. However, to date, we lack conditions guaranteeing that the dimension of the controllable subspace of a temporal network is larger than that of its static counterpart. In this work, we consider the case in which a static network is input connected but not controllable. We show that when the structure of the graph underlying the temporal network remains the same throughout each temporal snapshot while the (nonvanishing) edge weights vary, then the temporal network will be completely controllable almost always, even when its static counterpart is not. An upper bound on the number of snapshots needed to achieve controllability is also provided.