Poisson Tensor Research Articles

C-ziptf: stable tensor factorization for zero-inflated multi-dimensional genomics data.

In the past two decades, genomics has advanced significantly, with single-cell RNA-sequencing (scRNA-seq) marking a pivotal milestone. ScRNA-seq provides unparalleled insights into cellular diversity and has spurred diverse studies across multiple conditions and samples, resulting in an influx of complex multidimensional genomics data. This highlights the need for robust methodologies capable of handling the complexity and multidimensionality of such genomics data. Furthermore, single-cell data grapples with sparsity due to issues like low capture efficiency and dropout effects. Tensor factorizations (TF) have emerged as powerful tools to unravel the complex patterns from multi-dimensional genomics data. Classic TF methods, based on maximum likelihood estimation, struggle with zero-inflated count data, while the inherent stochasticity in TFs further complicates result interpretation and reproducibility. Our paper introduces Zero Inflated Poisson Tensor Factorization (ZIPTF), a novel method for high-dimensional zero-inflated count data factorization. We also present Consensus-ZIPTF (C-ZIPTF), merging ZIPTF with a consensus-based approach to address stochasticity. We evaluate our proposed methods on synthetic zero-inflated count data, simulated scRNA-seq data, and real multi-sample multi-condition scRNA-seq datasets. ZIPTF consistently outperforms baseline matrix and tensor factorization methods, displaying enhanced reconstruction accuracy for zero-inflated data. When dealing with high probabilities of excess zeros, ZIPTF achieves up to better accuracy. Moreover, C-ZIPTF notably enhances the factorization's consistency. When tested on synthetic and real scRNA-seq data, ZIPTF and C-ZIPTF consistently uncover known and biologically meaningful gene expression programs. Access our data and code at: https://github.com/klarman-cell-observatory/scBTF and https://github.com/klarman-cell-observatory/scbtf_experiments .

Simultaneous local normal forms of dynamical systems with singular underlying geometric structures

Abstract The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples ( X , G ) , where X is a dynamical system (vector field) and G is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where G is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where G is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure G is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and G are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when G and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.

Poisson tensor completion with transformed correlated total variation regularization

Tensor completion involves recovering the underlying tensor from partial observations, and in this paper we focus on the point that these observations obey the Poisson distribution. To contend with this problem, we adopt a popular method that minimizes the sum of the data-fitting term and the regularization term under a uniform sampling mechanism. Specifically, we consider the negative logarithmic maximum likelihood estimate of the Poisson distribution as the data-fitting term. To effectively characterize the intrinsic structure of the tensor data, we propose a parameter-free regularization term that can simultaneously capture the low rankness and local smoothness of the underlying tensor. Here, the transformed tensor nuclear norm is used to explore the low rankness under suitable unitary transformations. We present theoretical derivations to demonstrate the feasibility of the proposed model. Furthermore, we develop an algorithm based on the alternating direction multiplier method (ADMM) to efficiently solve the proposed optimization problem, with its overall convergence being established. A series of numerical experiments show that proposed model yields a pleasing accuracy over several state-of-the-art models.

A generalizable framework for low-rank tensor completion with numerical priors

Low-Rank Tensor Completion, a method which exploits the inherent structure of tensors, has been studied extensively as an effective approach to tensor completion. Whilst such methods attained great success, none have systematically considered exploiting the numerical priors of tensor elements. Ignoring numerical priors causes loss of important information regarding the data, and therefore prevents the algorithms from reaching optimal accuracy. Despite the existence of some individual works which consider ad hoc numerical priors for specific tasks, no generalizable frameworks for incorporating numerical priors have appeared. We present the Generalized CP Decomposition Tensor Completion (GCDTC) framework, the first generalizable framework for low-rank tensor completion that takes numerical priors of the data into account. We test GCDTC by further proposing the Smooth Poisson Tensor Completion (SPTC) algorithm, an instantiation of the GCDTC framework, whose performance exceeds current state-of-the-arts by considerable margins in the task of non-negative tensor completion, exemplifying GCDTC’s effectiveness. Our code is open-source.

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Brane Quantization of Toric Poisson Varieties

In this paper we propose a noncommutative generalization of the relationship between compact K\"ahler manifolds and complex projective algebraic varieties. Beginning with a prequantized K\"ahler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric K\"ahler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.

An optimal statistical and computational framework for generalized tensor estimation

This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of nonconvexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. Then we further consider a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data.

Structure-preserving reduced basis methods for Poisson systems

We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poisson manifold structure which encodes the physical properties, symmetries, and conservation laws of the dynamics. The goal is to design reduced basis methods for the general state-dependent degenerate Poisson structure based on a two-step approach. First, via a local approximation of the Poisson tensor, we split the Hamiltonian dynamics into an almost part and the trivial evolution of the Casimir invariants. Second, canonically symplectic reduced basis techniques are applied to the nontrivial component of the dynamics, preserving the local Poisson tensor kernel exactly. The global Poisson structure and the conservation properties of the phase flow are retained by the reduced model in the constant-valued case and up to errors in the Poisson tensor approximation in the state-dependent case. A priori error estimates for the solution of the reduced system are established. A set of numerical simulations is presented to corroborate the theoretical findings.

A decoupling property of some Poisson structures on Matn×d(C)×Matd×n(C) supporting GL(n,C)×GL(d,C) Poisson–Lie symmetry

We study a holomorphic Poisson structure defined on the linear space S(n,d)≔Matn×d(C)×Matd×n(C) that is covariant under the natural left actions of the standard GL(n,C) and GL(d,C) Poisson–Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the d = 1 special case gives a Poisson structure on S(n, 1), and we construct a local Poisson map from the Cartesian product of d independent copies of S(n, 1) into S(n, d), which is a holomorphic diffeomorphism in a neighborhood of 0. The Poisson structure on S(n, d) is the complexification of a real Poisson structure on Matn×d(C) constructed by the authors and Marshall, where a similar decoupling into d independent copies was observed. We also relate our construction to a Poisson structure on S(n, d) defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars–Schneider system by Hamiltonian reduction.

Variational auto-encoder based Bayesian Poisson tensor factorization for sparse and imbalanced count data

Non-negative tensor factorization models enable predictive analysis on count data. Among them, Bayesian Poisson-Gamma models can derive full posterior distributions of latent factors and are less sensitive to sparse count data. However, current inference methods for these Bayesian models adopt restricted update rules for the posterior parameters. They also fail to share the update information to better cope with the data sparsity. Moreover, these models are not endowed with a component that handles the imbalance in count data values. In this paper, we propose a novel variational auto-encoder framework called VAE-BPTF which addresses the above issues. It uses multi-layer perceptron networks to encode and share complex update information. The encoded information is then reweighted per data instance to penalize common data values before aggregated to compute the posterior parameters for the latent factors. Under synthetic data evaluation, VAE-BPTF tended to recover the right number of latent factors and posterior parameter values. It also outperformed current models in both reconstruction errors and latent factor (semantic) coherence across five real-world datasets. Furthermore, the latent factors inferred by VAE-BPTF are perceived to be meaningful and coherent under a qualitative analysis.

Deformation classes in generalized Kähler geometry

AbstractWe describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

Contravariant pseudo-Hessian manifolds and their associated Poisson structures

A contravariant pseudo-Hessian manifold is a manifold M endowed with a pair (∇,h) where ∇ is a flat connection and h is a symmetric bivector field satisfying a contravariant Codazzi equation. When h is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties similar to Poisson manifolds and, in fact, to any contravariant pseudo-Hessian manifold (M,∇,h) we associate naturally a Poisson tensor on TM. We investigate these properties and we study in details many classes of such structures in order to highlight the richness of the geometry of these manifolds.

U -duality extension of Drinfel’d double

Abstract A family of algebras $\mathcal{E}_n$ that extends the Lie algebra of the Drinfel’d double is proposed. This allows us to systematically construct the generalized frame fields $E_A{}^I$ which realize the proposed algebra by means of the generalized Lie derivative, i.e., . By construction, the generalized frame fields include a twist by a Nambu–Poisson tensor. A possible application to the non-Abelian extension of $U$-duality and a generalization of the Yang–Baxter deformation are also discussed.

On Riemann-Poisson Lie groups

A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in [4]. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.

Reduced Riemannian Poisson manifolds and Riemannian Poisson-Lie groups

Let (M,ΛM,〈,〉M) be a Poisson manifold equipped with a Riemannian metric 〈,〉M compatible with the Poisson structure ΛM and H a Lie group that acts on M properly, freely, by isometries and preserving the Poisson structure on M. There exists a unique Poisson structure ΛM/H and a unique Riemannian metric 〈,〉M/H on the reduced manifold M/H, generated by ΛM and 〈,〉M respectively. In this paper, we give necessary and sufficient conditions so that the compatibility conditions between the Poisson tensor ΛM and the metric 〈,〉M on M remain verified on the reduced Poisson manifold (M/H,ΛM/H,〈,〉M/H). When M=G is a Poisson-Lie group and H is a normal and closed coisotropic subgroup of G, we give interesting algebraic consequences associated with the compatibility between the Poisson tensor and the metric on G/H.

Invariant Nijenhuis Tensors and Integrable Geodesic Flows

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $\Pi_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $\Pi_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$.

Time-semantic-aware Poisson tensor factorization approach for scalable hotel recommendation

With the rapid development of tourism, personalized hotel recommendation has drawn growing attention from researchers, as online hotel booking becomes a challenging task when users confronted with a large volume of hotel information. However, most of conventional rating-based hotel recommendation approaches do not explicitly take into account the temporal behavior and review texts of users, which are essential to recommend hotels that match users’ preferences. In this paper, we propose a time-semantic-aware Poisson tensor factorization approach to learn comprehensively the temporal dynamics, multi-aspect ratings and review texts for hotel recommendation. This approach captures periodic effects of user activity with a time-aware factorization model. By exploiting the shared latent structure between users’ multi-aspect ratings and review texts, the approach has been shown to be able to predict more precisely the user’s preferences on hotels and easily be generalized to cold-start users for scalable hotel recommendation. To evaluate the performance of proposed method, experiments are conducted on two datasets collected from TripAdvisor. Experimental results of top-k hotel recommendation task reveal that the proposed model outperforms the state-of-the-art recommendation methods with a significant margin.

Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper 'Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation' by P. van der Kamp et al. [5], it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $ \ell(x,y) $, let $ B_1,B_2 $ be any two distinct points on the line $ \ell(x,y) = -c $, and let $ B_3,B_4 $ be any two distinct points on the line $ \ell(x,y) = c $. Set $ B_0 = \tfrac{1}{2}(B_1+B_3) $ and $ B_5 = \tfrac{1}{2}(B_2+B_4) $; these points lie on the line $ \ell(x,y) = 0 $. Finally, let $ B_\infty $ be the point at infinity on this line. Let $ \mathfrak E $ be the pencil of conics with the base points $ B_1,B_2,B_3,B_4 $. Then the composition of the $ B_\infty $-switch and of the $ B_0 $-switch on the pencil $ \mathfrak E $ is the Kahan discretization of a Hamiltonian vector field $ f = \ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix} $ with a quadratic Hamilton function $ H(x,y) $. This birational map $ \Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2 $ has three singular points $ B_0,B_2,B_4 $, while the inverse map $ \Phi_f^{-1} $ has three singular points $ B_1,B_3,B_5 $.

Queer Poisson brackets

We give a method to construct Poisson brackets $\{\cdot,\cdot\}$ on Banach manifolds~$M$, for which the value of $\{f,g\}$ at some point $m\in M$ may depend on higher order derivatives of the smooth functions $f,g\colon M\to{\mathbb R}$, and not only on the first-order derivatives, as it is the case on all finite-dimensional manifolds. We discuss specific examples in this connection, as well as the impact on the earlier research on Poisson geometry of Banach manifolds. Those brackets are counterexamples to the claim that the Leibniz property for any Poisson bracket on a Banach manifold would imply the existence of a Poisson tensor for that bracket.

Contravariant geometry and emergent gravity from noncommutative gauge theories

We investigate a relation of the contravariant geometry to the emergent gravity from noncommutative gauge theories. We give a refined formulation of the contravariant gravity, and provide solutions to the contravariant Einstein equation. We linearize the equation around background solutions, including curved ones. A noncommutative gauge theory on the Moyal plane can be rewritten as an ordinary gauge theory on a curved background via the Seiberg–Witten map; this is known as emergent gravity. We show that this phenomenon also occurs for a gauge theory on a noncommutative homogeneous Kähler background. We argue that the resulting geometry can be naturally described by the contravariant geometry under an identification of the fluctuation of the Poisson tensor with the field strength obtained by the Seiberg–Witten map. These results indicate that the contravariant gravity is a suitable framework for noncommutative spacetime physics.