Showing papers on "Basis (linear algebra) published in 2019"

Self-Supervised Spatio-Temporal Representation Learning for Videos by Predicting Motion and Appearance Statistics

Abstract: We address the problem of video representation learning without human-annotated labels. While previous efforts address the problem by designing novel self-supervised tasks using video data, the learned features are merely on a frame-by-frame basis, which are not applicable to many video analytic tasks where spatio-temporal features are prevailing. In this paper we propose a novel self-supervised approach to learn spatio-temporal features for video representation. Inspired by the success of two-stream approaches in video classification, we propose to learn visual features by regressing both motion and appearance statistics along spatial and temporal dimensions, given only the input video data. Specifically, we extract statistical concepts (fast-motion region and the corresponding dominant direction, spatio-temporal color diversity, dominant color, etc.) from simple patterns in both spatial and temporal domains. Unlike prior puzzles that are even hard for humans to solve, the proposed approach is consistent with human inherent visual habits and therefore easy to answer. We conduct extensive experiments with C3D to validate the effectiveness of our proposed approach. The experiments show that our approach can significantly improve the performance of C3D when applied to video classification tasks. Code is available at https://github.com/laura-wang/video_repres_mas.

An Analytic Approach to BCFT$_d$

Abstract: We develop an analytic approach to Boundary Conformal Field Theory (BCFT), focussing on the two-point function of a general pair of scalar primary operators. The resulting crossing equation can be thought of as a vector equation in an infinite-dimensional space $$ \mathcal{V} $$ of analytic functions of a single complex variable. We argue that in a unitary theory, functions in $$ \mathcal{V} $$ satisfy a boundedness condition in the Regge limit. We identify a useful basis for $$ \mathcal{V} $$ , consisting of bulk and boundary conformal blocks with scaling dimensions which appear in OPEs of the mean field theory correlator. Our main achievement is an explicit expression for the action of the dual basis (the basis of linear functionals on $$ \mathcal{V} $$ ) on an arbitrary conformal block. The practical merit of our basis is that it trivializes the study of perturbations around mean field theory. Our results are equivalent to a BCFT version of the Polyakov bootstrap. Our derivation of the expressions for the functionals relies on the identification of the Polyakov blocks with (suitably improved) boundary and bulk Witten exchange diagrams in AdSd+1. We also provide another conceptual perspective on the Polyakov block expansion and the associated functionals, by deriving a new Lorentzian OPE inversion formula for BCFT.

Feynman integrals and intersection theory

Abstract: We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.

Model Order Reduction for Problems with large Convection Effects

Abstract: The reduced basis method allows to propose accurate approximations for many parameter dependent partial differential equations, almost in real time, at least if the Kolmogorov n-width of the set of all solutions, under variation of the parameters, is small. The idea is that any solutions may be well approximated by the linear combination of some well chosen solutions that are computed offline once and for all (by another, more expensive, discretization) for some well chosen parameter values. In some cases, however, such as problems with large convection effects, the linear representation is not sufficient and, as a consequence, the set of solutions needs to be transformed/twisted so that the combination of the proper twist and the appropriate linear combination recovers an accurate approximation. This paper presents a simple approach towards this direction, preliminary simulations support this approach.

Darboux transformation and analytic solutions for a generalized super-NLS-mKdV equation

Abstract: Darboux transformation is an efficient method for solving different nonlinear partial differential equations. In this paper, on the basis of a Lie super-algebras, a generalized super-NLS-mKdV equation is solved by the Darboux transformation. The analytic solutions are presented with the help of symbolic computation. Besides, two special cases are given to make the solution intuitive. Dynamic properties of solitons are also discussed.

YBC: a stellar bolometric corrections database with variable extinction coefficients. Application to PARSEC isochrones

Abstract: We present the YBC database of stellar bolometric corrections, in which we homogenise widely used theoretical stellar spectral libraries and provide BCs for many popular photometric systems, including Gaia filters. The database can easily be extended to additional photometric systems and stellar spectral libraries. The web interface allows users to transform their catalogue of theoretical stellar parameters into magnitudes and colours of selected filter sets. The BC tables can be downloaded or implemented into large simulation projects using the interpolation code provided with the database. We computed extinction coefficients on a star-by-star basis, hence taking into account the effects of spectral type and non-linearity dependency on the total extinction. We illustrate the use of these BCs in PARSEC isochrones. We show that using spectral-type dependent extinction coefficients is necessary for Gaia filters whenever A V ≳ 0.5 mag. Bolometric correction tables for rotating stars and tables of limb-darkening coefficients are also provided.

Divide, Conquer and Combine: Hierarchical Feature Fusion Network with Local and Global Perspectives for Multimodal Affective Computing

Abstract: We propose a general strategy named ‘divide, conquer and combine’ for multimodal fusion. Instead of directly fusing features at holistic level, we conduct fusion hierarchically so that both local and global interactions are considered for a comprehensive interpretation of multimodal embeddings. In the ‘divide’ and ‘conquer’ stages, we conduct local fusion by exploring the interaction of a portion of the aligned feature vectors across various modalities lying within a sliding window, which ensures that each part of multimodal embeddings are explored sufficiently. On its basis, global fusion is conducted in the ‘combine’ stage to explore the interconnection across local interactions, via an Attentive Bi-directional Skip-connected LSTM that directly connects distant local interactions and integrates two levels of attention mechanism. In this way, local interactions can exchange information sufficiently and thus obtain an overall view of multimodal information. Our method achieves state-of-the-art performance on multimodal affective computing with higher efficiency.

Galerkin approximation of dynamical quantities using trajectory data

Abstract: Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to the estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system’s dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time.Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to the estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system’s dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time.

Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

Abstract: Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with $\widetilde{\cal O}(N^{3/2} \lambda)$ T complexity, where $N$ is number of orbitals and $\lambda$ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

Atomic Cluster Expansion: Completeness, Efficiency and Stability

Abstract: The Atomic Cluster Expansion (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation by proposing a precomputation algorithm that yields immediate guarantees that a complete basis is obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.

Frames and Numerical Approximation

Abstract: Functions of one or more variables are usually approximated with a basis: a complete, linearly independent system of functions that spans a suitable function space. The topic of this paper is the n...

Reduced-Order Modeling with Artificial Neurons for Gravitational-Wave Inference

Abstract: Gravitational-wave data analysis is rapidly absorbing techniques from deep learning, with a focus on convolutional networks and related methods that treat noisy time series as images. We pursue an alternative approach, in which waveforms are first represented as weighted sums over reduced bases (reduced-order modeling); we then train artificial neural networks to map gravitational-wave source parameters into basis coefficients. Statistical inference proceeds directly in coefficient space, where it is theoretically straightforward and computationally efficient. The neural networks also provide analytic waveform derivatives, which are useful for gradient-based sampling schemes. We demonstrate fast and accurate coefficient interpolation for the case of a four-dimensional binary-inspiral waveform family and discuss promising applications of our framework in parameter estimation.

A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets:

Abstract: The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wave...

Genuinely nonlocal product bases: Classification and entanglement-assisted discrimination

Abstract: An orthogonal product basis of a composite Hilbert space is genuinely nonlocal if the basis states are locally indistinguishable across every bipartition. From an operational point of view, such a basis corresponds to a separable measurement that cannot be implemented by local operations and classical communication unless all the parties come together in a single location. In this work we classify genuinely nonlocal product bases into different categories. Our classification is based on the state elimination property of the set via orthogonality-preserving measurements when all the parties are spatially separated or different subsets of the parties come together. We then study local state discrimination protocols for several such bases with additional entangled resources shared among the parties. Apart from consuming less entanglement than teleportation-based schemes, our protocols indicate operational significance of the proposed classification and exhibit nontrivial use of genuine entanglement in the local state discrimination problem.

Natural orbitals for ab initio no-core shell model calculations

Abstract: We explore the impact of optimizations of the single-particle basis on the convergence behavior and robustness of ab initio no-core shell model calculations Our focus is on novel basis sets defined by the natural orbitals of a correlated one-body density matrix that is obtained in second-order many-body perturbation theory Using a perturbatively improved density matrix as starting point informs the single-particle basis about the dominant correlation effects on the global structure of the many-body state, while keeping the computational cost for the basis optimization at a minimum Already the comparison of the radial single-particle wavefunctions reveals the superiority of the natural-orbital basis compared to a Hartree-Fock or harmonic oscillator basis, and it highlights pathologies of the Hartree-Fock basis We compare the model-space convergence of energies, root-mean-square radii, and selected electromagnetic observables with all three basis sets for selected p-shell nuclei using chiral interactions with explicit three-nucleon terms In all cases the natural-orbital basis provides the fastest and most robust convergence, making it the most efficient basis for no-core shell model calculations As an application we present no-core shell model calculations for the ground-state energies of all oxygen isotopes and assess the accuracy of the normal-ordered two-body approximation of the three-nucleon interaction in the natural-orbital basis

Decompositions of dependence for high-dimensional extremes

Abstract: Employing the framework of regular variation, we propose two decompositions which help to summarize and describel high-dimensional tail dependence. Via transformation, we define a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regularly varying random vectors preserve regular variation. Rather than model regular-variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. This matrix is positive semidefinite, and eigendecomposition allows one to interpret tail dependence via the resulting eigenbasis. Additionally this matrix is completely positive, and a resulting decomposition allows one to easily construct regularly varying random vectors which share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall data and financial return data.

Error-consistent segmented contracted all-electron relativistic basis sets of double- and triple-zeta quality for NMR shielding constants

Abstract: We present property-tailored all-electron relativistic Karlsruhe basis sets for the elements hydrogen to radon. The modifications described herein use at most four additional primitive basis functions and re-optimized contraction coefficients of the inner-most segment. Thus, the shielding constants are improved while maintaining the compactness of the basis set. A large set of 255 closed-shell molecules was used to assess the quality of the developed bases throughout the periodic table of elements.

Infinite Alphabet Edge Shift Spaces via Ultragraphs and Their C*-Algebras

Abstract: We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.

Qubitization of Arbitrary Basis Quantum Chemistry by Low Rank Factorization

Abstract: Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with $\widetilde{\cal O}(N^{3/2} \lambda)$ T complexity, where $N$ is number of orbitals and $\lambda$ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

Standard Model Effective Field Theory from On-shell Amplitudes

Abstract: We present a general method of constructing unfactorizable on-shell amplitudes (amplitude basis), and build up their one-to-one correspondence to the independent and complete operator basis in effective field theory (EFT). We apply our method to the Standard Model EFT, and identify the amplitude basis in dimension 5 and 6, which correspond to the Weinberg operator and operators in Warsaw basis except for some linear combinations.

Generalized Many-Body Expanded Full Configuration Interaction Theory

Abstract: Facilitated by a rigorous partitioning of a molecular system’s orbital basis into two fundamental subspaces—a reference and an expansion space, both with orbitals of unspecified occupancy—we genera...

Prescriptive unitarity for non-planar six-particle amplitudes at two loops

Abstract: We extend the applications of prescriptive unitarity beyond the planar limit to provide local, polylogarithmic, integrand-level representations of six-particle MHV scattering amplitudes in both maximally supersymmetric Yang-Mills theory and gravity. The integrand basis we construct is diagonalized on a spanning set of non-vanishing leading singularities that ensures the manifest matching of all soft-collinear singularities in both theories. As a consequence, this integrand basis naturally splits into infrared-finite and infrared-divergent parts, with hints toward an integrand-level exponentiation of infrared divergences. Importantly, we use the same basis of integrands for both theories, so that the presence or absence of residues at infinite loop momentum becomes a feature detectable by inspecting the cuts of the theory. Complete details of our results are provided as sup- plementary material.

Scan efficiency of structured illumination in iterative single pixel imaging

Abstract: Single-pixel imaging (SPI) is an innovative technique that images an object from non-pixelated detection. To do so, SPI has to conduct structured illumination that functions as a way to scan the object. The illumination basis and corresponding scanned intensities are then used for correlation measurement to reconstruct an image. In this process, the illumination structure, or scanning basis plays an important role on the scanning efficiency and therefore reconstruction quality. In this work we discuss the efficiency of different scanning basis in iterative SPI. A comparison between raster scan (RS) and multi-pixel structured scan (MS) in SPI is carried out under the criterions of signal to noise ratio and structural similarity index. Theoretical analysis is followed with demonstration from both experiment and simulation. Our conclusion is believed to be useful guidelines for choosing the right illumination basis depending on the iterative SPI application situation.

Deep learning: Extrapolation tool for ab initio nuclear theory

Abstract: Ab initio approaches in nuclear theory, such as the no-core shell model (NCSM), have been developed for approximately solving finite nuclei with realistic strong interactions. The NCSM and other approaches require an extrapolation of the results obtained in a finite basis space to the infinite basis space limit and assessment of the uncertainty of those extrapolations. Each observable requires a separate extrapolation and many observables have no proven extrapolation method. We propose a feed-forward artificial neural network (ANN) method as an extrapolation tool to obtain the ground-state energy and the ground-state point-proton root-mean-square (rms) radius along with their extrapolation uncertainties. The designed ANNs are sufficient to produce results for these two very different observables in $^{6}\mathrm{Li}$ from the ab initio NCSM results in small basis spaces that satisfy the following theoretical physics condition: independence of basis space parameters in the limit of extremely large matrices. Comparisons of the ANN results with other extrapolation methods are also provided.