Dimensional Representation Research Articles

Spatially aware dimension reduction for spatial transcriptomics

Spatial transcriptomics are a collection of genomic technologies that have enabled transcriptomic profiling on tissues with spatial localization information. Analyzing spatial transcriptomic data is computationally challenging, as the data collected from various spatial transcriptomic technologies are often noisy and display substantial spatial correlation across tissue locations. Here, we develop a spatially-aware dimension reduction method, SpatialPCA, that can extract a low dimensional representation of the spatial transcriptomics data with biological signal and preserved spatial correlation structure, thus unlocking many existing computational tools previously developed in single-cell RNAseq studies for tailored analysis of spatial transcriptomics. We illustrate the benefits of SpatialPCA for spatial domain detection and explores its utility for trajectory inference on the tissue and for high-resolution spatial map construction. In the real data applications, SpatialPCA identifies key molecular and immunological signatures in a detected tumor surrounding microenvironment, including a tertiary lymphoid structure that shapes the gradual transcriptomic transition during tumorigenesis and metastasis. In addition, SpatialPCA detects the past neuronal developmental history that underlies the current transcriptomic landscape across tissue locations in the cortex.

Hybrid Kronecker Product Decomposition and Approximation

Discovering underlying low dimensional structure of a high-dimensional matrix is traditionally done through low rank matrix approximations in the form of a sum of rank-one matrices. In this article, we propose a new approach. We assume a high-dimensional matrix can be approximated by a sum of a small number of Kronecker products of matrices with potentially different configurations, named as a hybrid Kronecker outer Product Approximation (hKoPA). It provides an extremely flexible way of dimension reduction compared to the low-rank matrix approximation. Challenges arise in estimating a hKoPA when the configurations of component Kronecker products are different or unknown. We propose an estimation procedure when the set of configurations are given, and a joint configuration determination and component estimation procedure when the configurations are unknown. Specifically, a least squares backfitting algorithm is used when the configurations are given. When the configurations are unknown, an iterative greedy algorithm is developed. Both simulation and real image examples show that the proposed algorithms have promising performances. Some identifiability conditions are also provided. The hybrid Kronecker product approximation may have potentially wider applications in low dimensional representation of high-dimensional data. Supplementary materials for this article are available online.

A Multigroup Homogeneous Flux Reconstruction Method Based on the ANOVA-HDMR Decomposition

In this paper, we formulate a new flux reconstruction method based on the High Dimensional Model Representation (HDMR) of the nodal flux shape. The method requires the conventional nodal parameters obtained as a result of the transverse-integration nodal solution procedure as well as corner values calculated by a combination of finite-volume and finite-difference approximations. The proposed flux reconstruction procedure is applicable to rectangular nodes and more than two energy groups. The flux reconstruction method is applied to a number of homogeneous and heterogeneous test problems in order to evaluate its accuracy.

Logical design of multi-model data warehouses

Multi-model DBMSs, which support different data models with a fully integrated backend, have been shown to be beneficial to data warehouses and OLAP systems. Indeed, they can store data according to the multidimensional model and, at the same time, let each of its elements be represented through the most appropriate model. An open challenge in this context is the lack of methods for logical design. Indeed, in a multi-model context, several alternatives emerge for the logical representation of dimensions and facts. The goal of this paper is to devise a set of guidelines for the logical design of multi-model data warehouses so that the designer can achieve the best trade-off between features such as querying, storage, and ETL. To this end, for each model considered (relational, document-based, and graph-based) and for each type of multidimensional element (e.g., non-strict hierarchy) we propose some solutions and carry out a set of intra-model and inter-model comparisons. The resulting guidelines are then tested on a case study that shows all types of multidimensional elements.

The product of lattice covolume and discrete series formal dimension: [formula omitted]-adic [formula omitted

Let F be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in PGL(2,F) and the formal dimension of a discrete series representation of GL(2,F). The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

Computation of invariant sets via immersion for discrete-time nonlinear systems

In this paper, we propose an approach for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we focus on the maximal admissible invariant set contained in some given constraint set. For special types of nonlinear systems, which can be exactly immersed into higher dimensional linear systems with state transformations, invariant sets of the original nonlinear system can be characterized using the higher dimensional linear representation. For general nonlinear systems without the immersibility property, approximate immersions are defined in a local region within some tolerance and linear approximations are computed by leveraging the fixed-point iteration technique for invariant sets. Given the bound on the mismatch between the linear approximation and the original system, we provide an invariant inner approximation of the maximal admissible invariant set by a tightening procedure.

A Fast CPU Real-Time Facial Expression Detector Using Sequential Attention Network for Human–Robot Interaction

Facial expression detection is a method to predict human facial emotions. This work is a trending research topic that can be implemented for human-robot interaction. More recently, deep convolutional neural network provides a robust extractor features but tends to be slow in real-time implementations and often requires a large memory and graphics processing units for fast execution. In this article, an efficient CPU-based facial expression detector is proposed using a sequential attention network to improve the baseline performance. The proposed attention network consists of three modules, global representation to capture the global features, channel representation, and dimension representation, which are focused on the channel and using spatial attention to discriminate local features. The efficient partial transfer module is also presented as a light backbone to extract facial features from an image. The entire module is trained and tested on several benchmarks to classify seven facial expressions. As a result, the proposed model reaches an accuracy of 98.18%, 98.75%, 95.63%, and 74.17% on CK+, JAFFE, KDEF, and FER-2013, respectively. It achieves competitive performance when compared to state-of-the-art methods. Lastly, it is integrated with a face detector and runs in real-time without a constraint at 69 frames per second on a CPU.

On the partition function of the Sp(4) integrable vertex model

In this paper we investigate certain fusion relations associated to an integrable vertex model on the square lattice which is invariant under Sp(4) symmetry. We establish a set of functional relations which include a transfer matrix inversion identity. The solution of these relations in the thermodynamic limit allows us to compute the partition function per site of the fundamental Sp(4) representation of the vertex model. As a byproduct we also obtain the partition function per site of a vertex model mixing the four and five dimensional representations of the Sp(4) symmetry.

Open-string integrals with multiple unintegrated punctures at genus one

We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the A-cycle of a torus. We construct a vector of such integrals which closes after taking a total differential with respect to the N unintegrated punctures and the modular parameter τ. These integrals are found to satisfy the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power series in α′ — the string length squared- in terms of elliptic multiple polylogarithms (eMPLs). In the N-puncture case, the KZB equation reveals a representation of B1,N, the braid group of N strands on a torus, acting on its solutions. We write the simplest of these braid group elements — the braiding one puncture around another — and obtain generating functions of analytic continuations of eMPLs. The KZB equations in the so-called universal case is written in terms of the genus-one Drinfeld-Kohno algebra mathfrak{t} 1,N ⋊ mathfrak{d} , a graded algebra. Our construction determines matrix representations of various dimensions for several generators of this algebra which respect its grading up to commuting terms.

Selective effects of a brain tumor on the metric representation of the hand: a pre- versus post-surgery comparison

Body representation disorders are complex, varied, striking, and very disabling in most cases. Deficits of body representation have been described after lesions to multimodal and sensorimotor cortical areas. A few studies have reported the effects of tumors on the representation of the body, but little is known about the changes after tumor resection. Moreover, the impact of brain lesions on the hand size representation has been investigated in few clinical cases. Hands are of special importance, as no other body part has the ability for movement and interaction with the environment that the hands have, and we use them for a multitude of daily activities. Studies with clinical population can add further knowledge into the way hands are represented. Here, we report a single case study of a patient (AM) who was an expert bodybuilder and underwent a surgery to remove a glioblastoma in the left posterior prefrontal and precentral cortex at the level of the hand’s motor region. Pre- (20 days) and post- (4 months) surgery assessment did not show any motor or cognitive impairments. A hand localization task was used, before and after surgery (12 months), to measure possible changes of the metric representation of his right hand. Results showed a post-surgery modulation of the typically distorted hand representation, with an overall accuracy improvement, especially on width dimension. These findings support the direct involvement of sensorimotor areas in the implicit representation of the body size and its relevance on defining specific size representation dimensions.

Searching for gravity without a metric

Recently it has been explicitly shown how a theory with global $GL(d,\mathbb{R})$ coordinate (affine) invariance which is spontaneously broken down to its Lorentz subgroup will have as its Goldstone fields enough degrees of freedom to create a metric and a covariant derivative arXiv:1105.5848. Such a theory would constitute an effective theory of gravity. So far however, no explicit theory has been found which exhibits this symmetry breaking pattern, mainly due to the difficulty of even writing down a $GL(d,\mathbb{R})$ invariant actions in the absence of a metric. In this paper we explicitly construct an affine generalization of the Dirac action employing infinite dimensional spinorial representations of the group. This implies that it is built from an infinite number of spinor Lorentz multiplets. We introduce a systematic procedure for obtaining $GL(d,\mathbb{R})$ invariant interaction terms to obtain quite general interacting models. Such models have order operators whose expectation value can break affine symmetry to Poincar\'{e} symmetry. We discuss possible interactions and mechanisms for this symmetry breaking to occur, which would provide a dynamical explanation of the Lorentzian signature of spacetime.

Dimension of representation and character varieties for two- and three-orbifolds

We consider varieties of representations and characters of 2 and 3-dimensional orbifolds in semisimple Lie groups, and we focus on computing their dimension. For hyperbolic 3-orbifolds, we consider the component of the variety of characters that contains the holonomy composed with the principal representation, we show that its dimension equals half the dimension of the variety of characters of the boundary. We also show that this is a lower bound for the dimension of generic components. We furthermore provide tools for computing dimensions of varieties of characters of 2-orbifolds, including the Hitchin component. We apply this computation to the dimension growth of varieties of characters of some 3-dimensional manifolds in SL(n,C).

Learning fair representations by separating the relevance of potential information

Representation learning has recently been used to remove sensitive information from data and improve the fairness of machine learning algorithms in social applications. However, previous works that used neural networks are opaque and poorly interpretable, as it is difficult to intuitively determine the independence between representations and sensitive information. The internal correlation among data features has not been fully discussed, and it may be the key to improving the interpretability of neural networks. A novel fair representation algorithm referred to as FRC is proposed from this conjecture. It indicates how representations independent of multiple sensitive attributes can be learned by applying specific correlation constraints on representation dimensions. Specifically, dimensions of the representation and sensitive attributes are treated as statistical variables. The representation variables are divided into two parts related to and unrelated to the sensitive variables by adjusting their absolute correlation coefficient with sensitive variables. The potential impact of sensitive information on representations is concentrated in the related part. The unrelated part of the representation can be used in downstream tasks to yield fair results. FRC takes the correlation between dimensions as the key to solving the problem of fair representation. Empirical results show that our representations enhance the ability of neural networks to show fairness and achieve better fairness-accuracy tradeoffs than state-of-the-art works.

Relations between cusp forms sharing Hecke eigenvalues

In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on G L n ( A F ) GL_n(\mathbb {A}_F) is contained in the set of Hecke eigenvalues of a cusp form on G L m ( A F ) GL_m(\mathbb {A}_F) for n ≤ m n \leq m . This question is closely related to a question about finite dimensional representations of an abstract group, which also we consider in this work.

Wave episode based Gaussian process regression for extreme event statistics in ship dynamics: Between the Scylla of Karhunen–Loève convergence and the Charybdis of transient features

For a number of structural and fatigue problems in marine science, engineers need accurate statistics for kinematic and dynamical quantities, such as displacements and bending moments. For many such problems the response statistics depend nonlinearly on input stochastic processes, such as sea surface elevation. Nonlinearity prevents the use of standard linear methods, leading to the adoption of expensive experiments and time-consuming numerical simulations. In order to avoid this cost we present a machine learning framework to minimize the training set requirements. This framework consists of two parts. First, we use the Karhunen–Loève theorem to represent stochastic sea states with finite-time wave episodes, which have low dimensionality that nonetheless captures the features important to hydrodynamics and structural mechanics. However, the choice of the wave episode duration is ‘caught’ between the Scylla of slow Karhunen–Loève series convergence for long time durations and the Charybdis of missing transient behaviors when the interval is short. To combat this dilemma, we propose a division into a region, designed for parametric interpolation and machine learning, and a stochastic prelude region, designed to probabilistically model transients. The second part of the framework consists of a Gaussian Process Regression (GPR) model designed to learn the mapping from wave episodes to structural outputs. GPR is able to take advantage of the low dimensional parametric representation of the sea state in order to converge with reasonably-sized training sets (on the order n≈100). At the same time, we use a low-dimensional representation in order to represent the stochastic response time series. The principal advantages of the Gaussian process surrogate are the blazing speed of evaluation – ten thousand times faster than the direct method – and the built in uncertainty quantification. Taken together, we can reconstruct the statistics of the responses by sampling sea states via the Karhunen–Loève construction, estimating the corresponding model outputs using the trained GPR, and estimating statistics of interest through Monte-Carlo calculation on the surrogate model. Finally the developed framework allows for trivial computation of the statistics for different sea spectra, i.e. without the need to retrain the Gaussian Process Regression scheme.

Generalized weighted permutation entropy.

A novel heuristic approach is proposed here for time series data analysis, dubbed Generalized weighted permutation entropy, which amalgamates and generalizes beyond their original scope two well established data analysis methods: Permutation entropy and Weighted permutation entropy. The method introduces a scaling parameter to discern the disorder and complexity of ordinal patterns with small and large fluctuations. Using this scaling parameter, the complexity-entropy causality plane is generalized to the complexity-entropy-scale causality box. Simulations conducted on synthetic series generated by stochastic, chaotic, and random processes, as well as real world data, are shown to produce unique signatures in this three dimensional representation.

Using frequency selectivity to examine category-informative dimension-selective attention

In everyday situations, listeners may selectively attend to acoustic dimensions (e.g., frequency) within complex sounds and ignore other simultaneous dimensions. Speech sound categories are defined over multiple dimensions that vary in informativeness; thus, speech perception may demand selective attention to diagnostic dimensions. Here, we examine how selective attention affects cortical representations of informative acoustic dimensions during category learning. Participants completed a five-day training regimen wherein they learned four novel non-speech categories to criterion (Obasih et al., in prep). Crucially, stimuli were designed such that for two categories, categorization required reliance on – and perhaps selective attention to – acoustic patterns within high-frequency bands; the other two categories were differentiated by acoustic patterns in low-frequency bands. After training, listeners completed an fMRI session using the same non-speech categorization task. Best frequency for each auditory cortex voxel was obtained using tonotopic mapping. We hypothesized that when categorization depended on distinctions in certain frequency ranges (e.g., high frequencies), listeners would selectively attend to those frequencies, resulting in greater recruitment of voxels preferring those frequencies and possibly attenuation of signal in voxels preferring other frequencies. We present preliminary data from our study, with the goal of clarifying the cortical mechanisms supporting dimension-based auditory selective attention.

City2vec: Urban knowledge discovery based on population mobile network

Due to the needs of social and economic development, population movements between cities often occur on a large scale. Spontaneous population movements between cities constitute a Mobile Network of enormous scale, and although a considerable amount of research has been conducted to parse this structure using algorithms related to complex networks, it is not possible to quantitatively describe the differences and similarities between different cities. We use graph embedding algorithm that extends the traditional complex network approach of one-dimensional cognition of cities to two-dimensional cognition. It could project city information into a high-dimensional mathematical space by learning the population flow relationships and computing a unique vector representation. We parsed the cell phone number data in about 80 million POIs in 334 cities across China provided by Tencent eMap, and constructed a city Mobile Network containing 2,662,596 directed edges that could effectively capture long-term, long-distance population migrations. The city embedding is actually a reduced dimensional representation of the mobile network, which can retain richer original information. Our research method can not only accomplish the tasks that traditional complex network analysis methods can do well, but also achieve higher dimensional cognition of cities (e.g., spatial relationships between cities and maturity of urban agglomerations), which is an effective complement to traditional graph theory methods. • Capturing long-term population movements based on electronic map phone number data. • Raising the level of awareness of the city. • Compute embedding representations of nodes in city networks. • City vectors contain geographic location and culture attribute.

Spectra of Bethe subalgebras of $$Y(\mathfrak {gl}_n)$$ in tame representations

We study the eigenproblem for Bethe subalgebras of the Yangian $Y(\mathfrak{gl}_n)$ in tame representations, i.e. in finite dimensional representations which admit Gelfand-Tsetlin bases. Namely, we prove that for any tensor product of skew modules $V=\otimes_{i=1}^k V_{\lambda_i \setminus \mu_i}(z_i)$ over the Yangian $Y(\mathfrak{gl}_n)$ with generic $z_i$'s, the family of Bethe subalgebras $B(X)$ with $X$ being a regular element of the maximal torus of $GL_n$ (or, more generally, with $X \in \overline{M_{0,n+2}}$) acts with a cyclic vector on $V$. Moreover, for $X$ in the real form of $\overline{M_{0,n+2}}$ which is the closure of regular unitary diagonal matrices we show, that the family of subalgebras $B(X)$ acts with simple spectrum on $\otimes_{i=1}^k V_{\lambda_i \setminus \mu_i}(z_i)$ for generic $z_i$'s where all $V_{\lambda_i \setminus \mu_i}(z_i)$ are Kirillov-Reshetikhin modules. In the subsequent paper we will use this to define a KR-crystal structure on the spectrum of a Bethe subalgebra on $V$.

PO095 / #613 STEREOELECTROENCEPHALOGRAPHY OF THE INSULAR CORTEX WITH INTRAOPERATIVE O-ARM. OUR EXPERIENCE AND RESULTS: E-POSTER VIEWING

Refractory epilepsy affects 30% of epileptic patients, with devastating consequences for their personal and professional development. Of those, about 40% could benefit from surgical treatment, provided there is an identifiable accessible epileptic focus. When non-invasive diagnosis is inconclusive, further study is warranted by incursive registration of electrical brain activity with either subdural, epidural or deep electrodes. The latter, known as stereoelectroencephalography (SEEG), displays a three dimensional representation of the epileptic zone and permits identification of subcortical aberrancies. Traditional frame-based or the novel robotic-based stereotactic placement of deep electrodes, though very precise, are cumbersome and expensive. Thus, we propose a frameless neuronavigation technique, combined with intraoperative imaging, as a reliable and straightforward alternative. We present our experience and results with an intraoperative 3D fluoroscopic device and an articulated passive arm for SEEG electrode placement.