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We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical theory of Vietoris-Rips complexes, and for $p=1,$ this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the "$\ell_p$-Vietoris-Rips spaces" are homotopy equivalent to the manifold; (3) we demonstrate that the $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on $p$; and that the homology groups of the $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.

Both magnetism and spin-orbit coupling in systems with broken inversion symmetry lift the spin degeneracy of electronic bands, but the consequences of interplay between these mechanisms remain poorly understood. Here, we show that ultrathin transition ferromagnet-heavy metal bilayers exhibit anomalous temperature- and electric bias-dependent behaviors in the vicinity of the Curie temperature, inconsistent with the usual Weiss magnetism. Characterization by several complementary techniques and analysis of the dependence on composition reveal that these effects originate from interfacial spin-orbit interaction, which results in the emergence of a state with distinct magnetic and magnetoelectronic properties that can be described as Rashba magnetism. Our findings open a new route for the characterization and control of spin-orbit phenomena in heterostructures enabling the development of efficient spin-orbitronic devices., Comment: 6 pages, 4 figures, comments are welcome

For a digraph $G$ without multisquares and a field $\mathbb{F}$, we construct a basis of the vector space of path $n$-chains $\Omega_n(G;\mathbb{F})$ for $n\geq 0$, generalising the basis of $\Omega_3(G;\mathbb{F})$ constructed by Grigory'an. For a field $\mathbb{F},$ we consider the $\mathbb{F}$-path Euler characteristic $\chi^\mathbb{F}(G)$ of a digraph $G$ defined as the alternating sum of dimensions of path homology groups with coefficients in $\mathbb{F}.$ If $\Omega_\bullet(G;\mathbb{F})$ is a bounded chain complex, the constructed bases can be applied to compute $\chi^\mathbb{F}(G)$. We provide an explicit example of a digraph $\mathcal{G}$ whose $\mathbb{F}$-path Euler characteristic depends on whether the characteristic of $\mathbb{F}$ is two, revealing the differences between GLMY theory and the homology theory of spaces. This allows us to prove that there is no topological space $X$ whose homology is isomorphic to path homology of the digraph $H_*(X;\mathbb{K})\cong {\rm PH}_*(\mathcal{G};\mathbb{K})$ simultaneously for $\mathbb{K}=\mathbb{Z}$ and $\mathbb{K}=\mathbb{Z}/2\mathbb{Z}.$

SrIrO$_3$ is a metallic complex oxide with unusual electronic and magnetic properties believed to originate from electron correlations due to its proximity to Mott metal-insulator transition. However, the nature of its electronic state and the mechanism of metallic conduction remain poorly understood. We demonstrate that shot noise produced by nanoscale SrIrO$_3$ junctions is strongly suppressed, inconsistent with diffusive quasiparticle transport. Analysis of thermal effects and scaling with the junction length reveals that conduction is mediated by collective hopping of electrons almost localized by correlations. Our results provide insight into the non-Fermi liquid state close to Mott transition, and advance shot noise measurements as a powerful technique for the studies of quantum materials., Comment: Comments are welcome

The article is devoted to the study of diagonal digraphs, i.e. digraphs whose magnitude homology is concentrated on the diagonal. For a digraph $G$ and $\ell\geq 2,$ we describe a condition, denoted by $(\mathcal{V}_\ell)$, equivalent to vanishing of the second magnitude homology group ${\rm MH}_{2,k}(G,\mathbb{Z})$ for all $k>\ell.$ In particular, diagonal digraphs satisfy $(\mathcal{V}_2).$ As a corollary we obtain that the 2-dimensional CW-complex obtained from a diagonal undirected graph by attaching 2-cells to all squares and triangles of the graph is simply connected. We also give an interpretation of diagonality in terms of Koszul algebras: a digraph $G$ is diagonal if and only if the distance algebra $\sigma G$ is Koszul for any ground field; and if and only if $G$ satisfies $(\mathcal{V}_2)$ and the path cochain algebra $\Omega^\bullet(G)$ is Koszul for any ground field. To obtain a source of examples of digraphs, we study the extended Hasse diagram $\hat G_K$ of a pure simplicial complex $K$. For a triangulation $K$ of a topological manifold $M,$ we express the non-diagonal part of the magnitude homology of $\hat G_K$ via the homology of $M$. As a corollary we obtain that, if $K$ is a triangulation of a closed manifold $M$, then $\hat G_K$ is diagonal if and only if $M$ is a homology sphere.

In this paper, we study operations on functors in the category of abelian groups simplar to the derivation in the sense of Dold-Puppe. They are defined as derived limits of a functor applied to the relation subgroup over a category of free presentations of the group. The integral homology of the Eilenberg-Maclane space $K(\mathbb Z,3)$ appears as a part of description of these operations applied to symmetric powers.

We prove that the magnitude (co)homology of an enriched category can, under some technical assumptions, be described in terms of derived functors between certain abelian categories. We show how this statement is specified for the cases of quasimetric spaces, finite quasimetric spaces, and finite digraphs. For quasimetric spaces, we define the notion of a distance module over a quasimetric space, define the functor of (co)invariants of a distance module and show that the magnitude (co)homology can be presented via its derived functors. As a corollary we obtain that the magnitude cohomology of a quasimetric space can be presented in terms of Ext functors in the category of distance modules. For finite quasimetric spaces, we show that magnitude (co)homology can be presented in terms of Tor and Ext functors over a certain graded algebra, called the distance algebra of the quasimetric space. For finite digraphs, the distance algebra is a bound quiver algebra. In addition, we show that the magnitude cohomology algebra of a finite quasimetric space can be described as a Yoneda algebra.

Let $\Delta$ be the Dirichlet Laplacian on the interval $(0,\pi)$. The null controllability properties of the equation $$u_{tt}+\Delta^2 u+\rho (\Delta)^\alpha u_t=F(x,t)$$ are studied. Let $T>0$, and assume initial conditions $(u^0,u^1)\in Dom(\Delta)\times L^2(0,\pi)$. We first prove finite dimensional null control results: suppose $F(x,t)=f^1(t)h^1(x)+f^2(t)h^2(x)$ with $h^1,h^2$ given functions. For $\alpha \in [0,3/2)$, we prove that there exist $h^1,h^2\in L^2(0,\pi)$ such that for any $(u^0,u^1)$, there exist $L^2$ null controls $(f^1,f^2).$ For $\alpha< 1$ and $\rho <2$, we prove null controllability with $f^2=0$ and $h^1$ belonging to a large class of functions. For $\alpha\in [3/2,2)$, we prove spectral and null controllability both generally fail, but two dimensional weak controllability holds. Our second set of results pertains to $F(x,t)=\chi_\Omega(x)f(x,t)$, with $\Omega$ any open subset of $(0,\pi)$. For any $\alpha \in [0,3/2),$ we prove there exists a null control $f\in L^2(\Omega\times(0,T))$ To prove our main results, we use the Fourier method to rewrite the control problems as moment problems. These are then solved by constructing biorthogonal sets to the associated exponential families. These constructions seem to be non-standard and may be of independent interest.

For a real $r\geq 0,$ we consider the notion of $r$-homotopy equivalence in the category quasimetric spaces, which includes metric spaces and directed graphs. We show that for a finite quasimetric space $X$ there is a unique (up to isometry) $r$-homotopy equivalent quasimetric space of the minimal possible cardinality. It is called the $r$-minimal model of $X$. We use this to construct a decomposition of the magnitude-path spectral sequence of a digraph into a direct sum of spectral sequences with certain properties. We also construct an $r$-homotopy invariant ${\rm SH}^r_{n,I}(X)$ of a quasimetric space $X,$ called spectral homology, that generalizes many other invariants: the pages of the magnitude-path spectral sequence, including path homology, magnitude homology, blurred magnitude homology and reachability homology.

We assume that $M_0$ is a $d$-dimensional $C^{2,1}$-smooth submanifold of $R^n$. Let $K_0$ be the convex hull of $M_0,$ and $B^n_1(0)$ be the unit ball. We assume that $ M_0 \subseteq \partial K_0 \subseteq B^n_1(0).$ We also suppose that $M_0$ has volume ($d$-dimensional Hausdorff measure) less or equal to $V$, reach (i.e., normal injectivity radius) greater or equal to $\tau$. Moreover, we assume that $M_0$ is $R$-exposed, that is, tangent to every point $x \in M$ there is a closed ball of radius $R$ that contains $M$. Let $x_1, \dots, x_N$ be independent random variables sampled from uniform distribution on $M_0$ and $\zeta_1, \dots, \zeta_N$ be a sequence of i.i.d Gaussian random variables in $R^n$ that are independent of $x_1, \dots, x_N$ and have mean zero and covariance $\sigma^2 I_n.$ We assume that we are given the noisy sample points $y_i$, given by $$ y_i = x_i + \zeta_i,\quad \hbox{ for }i = 1, 2, \dots,N. $$ Let $\epsilon,\eta>0$ be real numbers and $k\geq 2$. Given points $y_i$, $i=1,2,\dots,N$, we produce a $C^k$-smooth function which zero set is a manifold $M_{rec}\subseteq R^n$ such that the Hausdorff distance between $M_{rec}$ and $M_0$ is at most $ \epsilon$ and $M_{rec}$ has reach that is bounded below by $c\tau/d^6$ with probability at least $1 - \eta.$ Assuming $d < c \sqrt{\log \log n}$ and all the other parameters are positive constants independent of $n$, the number of the needed arithmetic operations is polynomial in $n$. In the present work, we allow the noise magnitude $\sigma$ to be an arbitrarily large constant, thus overcoming a drawback of previous work.

We prove the following local version of Kakutani's characterization of ellipsoids: Let $V$ be a finite-dimensional real vector space, $B\subset V$ a convex body with 0 in its interior, and ${2\le k<\dim V}$ an integer. Suppose that the body $B$ is contained in a cylinder based on the cross-section $B \cap X$ for any $k$-plane $X$ from a connected open set of linear $k$-planes in $V$. Then in the region of $V$ swept by these $k$-planes $B$ coincides with either an ellipsoid, or a cylinder over an ellipsoid, or a cylinder over a $k$-dimensional base. For $k=2$ and $k=3$ we obtain as a corollary a local solution to Banach's isometric subspace problem: If all cross-sections of $B$ by $k$-planes from a connected open set are linearly equivalent, then the same conclusion as above holds., Comment: 16 pages

We give a necessary and sufficient condition for an inverse sequence $S_0 \leftarrow S_1 \leftarrow \dots$ indexed by natural numbers to have ${\rm lim}^1S=0$. This condition can be treated as a transfinite version of the Mittag-Leffler condition. We consider inverse sequences in an arbitrary abelian category having a generator and satisfying Grothendieck axioms ${\rm (AB3)}$ and ${\rm (AB4^*)}.$ We also show that the class of inverse sequences $S$ such that ${\rm lim}\: S={\rm lim}^1 S=0$ is the least class of inverse sequences containing the trivial inverse sequence and closed with respect to small limits and a certain type of extensions.

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions that we consider are questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, there is no a natural non-trivial way to define an abelian subgroup of a group, or a perfect quotient group of a group. As an auxiliary result we prove that, for any non-trivial subfunctor $F$ of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of $F.$ The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the $\infty$-category of spaces.

Composite quasi-particles with emergent functionalities in spintronic and quantum information science can be realized in correlated materials due to entangled charge, spin, orbital, and lattice degrees of freedom. Here we show that by reducing the lateral dimension of correlated antiferromagnet NiPS3 flakes to tens of nanometers, we can switch-off the bulk spin-orbit entangled exciton in the near-infrared (1.47 eV) and activate visible-range (1.8 to 2.2 eV) transitions with charge-transfer character. These ultra-sharp lines (<120 ueV at 4.2 K) share the spin-correlated nature of the bulk exciton by displaying a Neel temperature dependent linear polarization. Furthermore, exciton photoluminescence lineshape analysis reveals a polaronic character via coupling with at-least 3 phonon modes and a comb-like Stark effect through discretization of charges in each layer. These findings augment the knowledge on the many-body nature of excitonic quasi-particles in correlated antiferromagnets and also establish the nanoscale platform as promising for maturing integrated magneto-optic devices.

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.

We consider the terminal monad among those preserving the objects of a subcategory, and in particular preserving the image of a monad. Several common monads are shown to be uniquely characterized by the property of being terminal objects in the category of co-augmented endo-functors. Once extended to infinity categories, this gives, for example, a complete characterization of the well-known Bousfield-Kan R-homology completion. In addition, we note that an idempotent pro-completion tower can be associated with any co-augmented endo functor M, whose limit is the terminal monad that preserves the closure of ImM, the image of M, under finite limits. We conclude that some basic properties of the homological completion tower of a space can be formulated and proved for general monads over any category with limits, and characterized as universal

Gradient Boosted Decision Trees (GBDTs) are dominant machine learning algorithms for modeling discrete or tabular data. Unlike neural networks with millions of trainable parameters, GBDTs optimize loss function in an additive manner and have a single trainable parameter per leaf, which makes it easy to apply high-order optimization of the loss function. In this paper, we introduce high-order optimization for GBDTs based on numerical optimization theory which allows us to construct trees based on high-order derivatives of a given loss function. In the experiments, we show that high-order optimization has faster per-iteration convergence that leads to reduced running time. Our solution can be easily parallelized and run on GPUs with little overhead on the code. Finally, we discuss future potential improvements such as automatic differentiation of arbitrary loss function and combination of GBDTs with neural networks., Comment: NeurIPS 2022 Workshop: Order up! The Benefits of Higher-Order Optimization in Machine Learning

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square-commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.

We develop a two-orbital Hubbard model of electron correlations in ultrathin (111)-oriented fcc films of late transition metals such as Co and Ni. Our model indicates that the Mott-Hund's interaction results in ferromagnetic nearest-neighbor orbital correlations. Frustration associated with the mismatch between orbital and crystal symmetries prevents orbital ordering, resulting in the orbital liquid state. This state can be manifested in phenomena involving spin-orbit coupling, such as magnetic anisotropy., Comment: Manuscript published in Phys. Rev. Materials

B.\,Ya.\,Levin has proved that zero set of a sine type function can be presented as a union of a finite number of separated sets, that is an important result in the theory of exponential Riesz bases. In the present paper we extend Levin's result to a more general class of entire functions $F(z)$ with zeros in a strip $0

In today's world, the presence of online disinformation and propaganda is more widespread than ever. Independent publishers are funded mostly via digital advertising, which is unfortunately also the case for those publishing disinformation content. The question of how to remove such publishers from advertising inventory has long been ignored, despite the negative impact on the open internet. In this work, we make the first step towards quickly detecting and red-flagging websites that potentially manipulate the public with disinformation. We build a machine learning model based on multilingual text embeddings that first determines whether the page mentions a topic of interest, then estimates the likelihood of the content being malicious, creating a shortlist of publishers that will be reviewed by human experts. Our system empowers internal teams to proactively, rather than defensively, blacklist unsafe content, thus protecting the reputation of the advertisement provider., Comment: Disinformation Countermeasures and Machine Learning Workshop at ICML 2022

The problem of out-of-distribution detection for graph classification is far from being solved. The existing models tend to be overconfident about OOD examples or completely ignore the detection task. In this work, we consider this problem from the uncertainty estimation perspective and perform the comparison of several recently proposed methods. In our experiment, we find that there is no universal approach for OOD detection, and it is important to consider both graph representations and predictive categorical distribution., Comment: ICML 2022 PODS Workshop

We show that the mean-field resonating valence bond approximation proposed in 1987 by Baskaran, Zou, and Anderson describes gapless charge pair excitations confined to the boundaries of the spinon Brillouin zone. The existence of such pairs accounts for all the essential anomalous electronic properties of cuprates, with superconductivity arising due to the charge drag by the spinon superflow. This mechanism may be relevant to other unconventional superconductors., Comment: The submissions of this manuscript to Phys Rev., J. Phys, PSS, Physica C were rejected, this is its only publication record

Prediction of protein-ligand (PL) binding affinity remains the key to drug discovery. Popular approaches in recent years involve graph neural networks (GNNs), which are used to learn the topology and geometry of PL complexes. However, GNNs are computationally heavy and have poor scalability to graph sizes. On the other hand, traditional machine learning (ML) approaches, such as gradient-boosted decision trees (GBDTs), are lightweight yet extremely efficient for tabular data. We propose to use PL interaction features along with PL graph-level features in GBDT. We show that this combination outperforms the existing solutions.

We prove that, if all intersections of a convex body $B\subset\mathbb R^4$ with 3-dimensional linear subspaces are linearly equivalent,then $B$ is a centered ellipsoid. This settles the case $n=3$ of an old problem by Banach: Let $V$ be a normed vector space whose $n$-dimesional linear subspaces, for a fixed $n<\dim V$, are all isometric. Is $V$ necessarily an inner product space? The dimensions $n=3$ and $\dim V=4$ are the first case where the answer was not known, and in this case global topological methods do not help. Our proof is based on a differential geometric approach., Comment: 25 pages

We consider how a closed Riemannian manifold and its metric tensor can be approximately reconstructed from local distance measurements. In the part 1 of the paper, we considered the construction of a smooth manifold in the case when one is given the noisy distances $\tilde d(x,y)=d(x,y)+\varepsilon_{x,y}$ for all points $x,y\in X$, where $X$ is a $\delta$-dense subset of $M$ and $|\varepsilon_{x,y}|<\delta$. In this paper we consider a similar problem with partial data, that is, the approximate construction of the manifold $(M,g)$ when we are given $\tilde d(x,y)$ for $x\in X$ and $y \in U\cap X$, where $U$ is an open subset of $M$. As an application, we consider the inverse problem of determining the manifold $(M,g)$ with non-negative Ricci curvature from noisy observations of the heat kernel $G(y,z,t)$ with separated observation points $y\in U$ and source points $z\in M\setminus \overline U$ on the time interval $0

We prove that for a subring $R\subseteq \mathbb Q$ and a free group $F$ of rank at least $2$ the length of the Bousfield's $HR$-localization tower for $F$ is at least $\omega+\omega$. The key ingredient of the proof is the theory of polynomial functors over $\mathbb Q.$

We give an overview of five rationalization theories for spaces (Bousfield-Kan's $\mathbb Q$-completion; Sullivan's rationalization; Bousfield's homology rationalization; Casacuberta-Peschke's $\Omega$-rationalization; G\'{o}mez-Tato-Halperin-Tanr\'{e}'s $\pi_1$-fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups ($\mathbb Q$-completion; $H\mathbb Q$-localization; Baumslag rationalization) that extend the classical Malcev completion.

Using the functor of Baumslag rationalization of groups we construct a functor on the category of all (non necessarily simply connected) spaces that extends the classical rationalization of simply connected spaces. We study this functor and compare it with other extensions of the classical rationalization: Bousfield-Kan $\mathbb Q$-completion; Bousfield's homology rationalization; G\'{o}mez-Tato--Halperin--Tantr\'{e}'s $\pi_1$-fiberwise rationalization; and the localization with respect to the maps $n:S^1\to S^1$ that we call $\Omega$-rationalization., Comment: Results of this paper are not new. They can be found in https://link.springer.com/article/10.1007/BF02802510

Hexaferrite materials are highly demanded to develop and manufacture electronic devices operating at radio- and microwave frequencies. In the light of the prospects for their use in the forthcoming terahertz electronics, here, we present our results on the terahertz and infrared dielectric response of a typical representative of hexaferrites family, lead-substituted M-type barium hexaferrite doped with aluminum, Ba0.2Pb0.8AlxFe12-xO19, x(Al)=0.0, 3.0, and 3.3. We studied uniquely large and high-quality single crystals of the. Systematic and detailed investigations of the dependences of terahertz-infrared (frequencies 8 - 8000 cm-1) spectra of complex dielectric permittivity on the temperature, 4 - 300 K, and on the chemical composition, x(Al)=0.0, 1.2, 3.0, 3.3, were performed for polarizations of the electric field E-vector of the probing radiation normal and parallel to the crystallographic c-axis. A number of resonance absorption bands are discovered at infrared-terahertz frequencies and assigned to polar phonons and transitions between energy levels of the fine-structured ground state of Fe2+ (5E) ions. In contrast to undoped BaFe12O19, no softening of the lowest frequency A2u phonon is observed, indicating suppression of a displacive phase transition in substituted compounds. Basing on dielectric data and detailed X-ray experiments, we find that for all concentrations of Al3+ ions, x(Al)=0.0, 1.2, 3.0, and 3.3, they mainly occupy the 2a and 12k octahedral site positions and that the degree of substitution of iron in tetrahedral positions is not substantial. Along with fundamental findings, the obtained data on broad-band dielectric properties of Ba0.2Pb0.8AlxFe12-xO19 crystals provides the information that can be used for development and manufacture of electronic devices with operating frequencies lying in the terahertz spectral band.

For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.

The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces -- simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it.

We prove that for the free group of rank two $F$ the cokernel of the homomorphism to its Baumslag rationalization $F\to {\sf Bau}(F)$ is not abelian. Moreover, this cokernel contains a free subgroup of countable rank. This answers a question of Emmanuel Farjoun.

We experimentally demonstrate a proof-of-principle implementation of an almost ideal memristor - a two-terminal circuit element whose resistance is approximately proportional to the integral of the input signal over time. The demonstrated device is based on a thin-film ferromagnet/antiferromagnet bilayer, where magnetic frustration results in viscous magnetization dynamics enabling memristive functionality, while the external magnetic field plays the role of the driving input. The demonstrated memristor concept is amenable to downscaling and can be adapted for electronic driving, making it attractive for applications in neuromorphic circuits., Comment: 5 pages, 4 figures

Graph neural networks (GNNs) are powerful models that have been successful in various graph representation learning tasks. Whereas gradient boosted decision trees (GBDT) often outperform other machine learning methods when faced with heterogeneous tabular data. But what approach should be used for graphs with tabular node features? Previous GNN models have mostly focused on networks with homogeneous sparse features and, as we show, are suboptimal in the heterogeneous setting. In this work, we propose a novel architecture that trains GBDT and GNN jointly to get the best of both worlds: the GBDT model deals with heterogeneous features, while GNN accounts for the graph structure. Our model benefits from end-to-end optimization by allowing new trees to fit the gradient updates of GNN. With an extensive experimental comparison to the leading GBDT and GNN models, we demonstrate a significant increase in performance on a variety of graphs with tabular features. The code is available: https://github.com/nd7141/bgnn., Comment: ICLR 2021: camera-ready

We consider a convolution-type operator on vector bundles over metric-measure spaces. This operator extends the analogous convolution Laplacian on functions in our earlier work to vector bundles, and is a natural extension of the graph connection Laplacian. We prove that for Euclidean or Hermitian connections on closed Riemannian manifolds, the spectrum of this operator and that of the graph connection Laplacian both approximate the spectrum of the connection Laplacian., Comment: journal version, revised Lemma 4.1, added references

In Gel'fand's inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold $M$ with boundary from the knowledge of the boundary $\partial M,$ the Neumann eigenvalues $\lambda_j$ and the boundary values of the eigenfunctions $\varphi_j|_{\partial M}$. We show that this problem has a stable solution with quantitative stability estimates in a class of manifolds with bounded geometry. More precisely, we show that finitely many eigenvalues and the boundary values of corresponding eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the Gromov-Hausdorff sense. We provide an algorithm to construct this metric space. This result is based on an explicit estimate on the stability of the unique continuation for the wave operator., Comment: introduction updated, to appear in Anal. PDE

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $\mathbb Z,$ and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module $M$ over a principal ideal domain that connects the exterior and the symmetric powers $0\to \Lambda^n M\to M \otimes \Lambda^{n-1} M \to \dots \to S^{n-1}M \otimes M \to S^nM\to 0 $ is purely acyclic.

Given any symplectomorphism on $D^{2n} (n\geq 1)$ which is $C^{\infty}$ close to the identity, and any completely integrable Hamiltonian system $\Phi^t_H$ in the proper dimension, we construct a $C^{\infty}$ perturbation of $H$ such that the resulting Hamiltonian flow contains a "local Poincar\'{e} section" that "realizes" the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy non-expansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure)., Comment: Minor modification. arXiv admin note: text overlap with arXiv:2009.10143