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Agricultural landscapes are quite complex, especially in the Global South where fields are smaller, and agricultural practices are more varied. In this paper we report on our progress in digitizing the agricultural landscape (natural and man-made) in our study region of India. We use high resolution imagery and a UNet style segmentation model to generate the first of its kind national-scale multi-class panoptic segmentation output. Through this work we have been able to identify individual fields across 151.7M hectares, and delineating key features such as water resources and vegetation. We share how this output was validated by our team and externally by downstream users, including some sample use cases that can lead to targeted data driven decision making. We believe this dataset will contribute towards digitizing agriculture by generating the foundational baselayer., Comment: 34 pages, 7 tables, 15 figs

We propose a new model of a distributed game, called an ATS game, which is played on a non-deterministic asynchronous transition system -- a natural distributed finite-state device working on Mazurkiewicz traces. This new partial-information game is played between an environment and a distributed system comprising multiple processes. A distributed strategy uses causal past to make the next move. The key algorithmic question is to solve the game, that is, to decide the existence of a distributed winning strategy. It turns out ATS games are equivalent to asynchronous games, which are known to be undecidable. We prove that ATS games are undecidable in this article., Comment: 1 lemma 10 pages

We propose a local, past-oriented fragment of propositional dynamic logic to reason about concurrent scenarios modelled as Mazurkiewicz traces, and prove it to be expressively complete with respect to regular trace languages. Because of locality, specifications in this logic are efficiently translated into asynchronous automata, in a way that reflects the structure of formulas. In particular, we obtain a new proof of Zielonka's fundamental theorem and we prove that any regular trace language can be implemented by a cascade product of localized asynchronous automata, which essentially operate on a single process. These results refine earlier results by Adsul et al. which involved a larger fragment of past propositional dynamic logic and used Mukund and Sohoni's gossip automaton. Our new results avoid using this automaton, or Zielonka's timestamping mechanism and, in particular, they show how to implement a gossip automaton as a cascade product., Comment: 13 pages. Accepted for publication at LICS 2024

In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed non-trivial character. We study points $y,z\in V$ where (i) $z$ is obtained as the leading term of the action of a 1-parameter subgroup $\lambda (t)\subseteq G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H \subseteq G$. Let $O(z)$ (resp. $O(y)$) denote the $G$-orbits of $z$ (resp. $y$), and $\overline{O(z)}$ (resp. $\overline{O(y)}$) their closures, then (i) implies that $z\in \overline{O(y)}$. We address the question: under what conditions can (i) and (ii) be simultaneously satisfied, i.e, there exists a 1-PS $\lambda \subseteq G$ for which $z$ is observed as a limit of $y$. Using $\lambda$, we develop a leading term analysis which applies to $V$ as well as to ${\cal G}= Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\cal K}$ and ${\cal H}$, the Lie algebras of $K$ and $H$ respectively. Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal H}$ which connects $y$ and $z$ through their Lie algebras. We develop the properties of $\hat{\cal K}$ and relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the normal slice to the orbit $O(z)$. We examine the case of {\em alignment} when a semisimple element belongs to both ${\cal H}$ and ${\cal K}$, and the conditions for the same. We illustrate some consequences of alignment. Next, we examine the possibility of {\em intermediate $G$-varieties} $W$ which lie between the orbit closures of $z$ and $y$, i.e. $\overline{O(z)} \subsetneq W \subsetneq O(y)$. These have a direct bearing on representation theoretic as well as geometric properties which connect $z$ and $y$.

Ubiquitous sensing from wearable devices in the wild holds promise for enhancing human well-being, from diagnosing clinical conditions and measuring stress to building adaptive health promoting scaffolds. But the large volumes of data therein across heterogeneous contexts pose challenges for conventional supervised learning approaches. Representation Learning from biological signals is an emerging realm catalyzed by the recent advances in computational modeling and the abundance of publicly shared databases. The electrocardiogram (ECG) is the primary researched modality in this context, with applications in health monitoring, stress and affect estimation. Yet, most studies are limited by small-scale controlled data collection and over-parameterized architecture choices. We introduce \textbf{WildECG}, a pre-trained state-space model for representation learning from ECG signals. We train this model in a self-supervised manner with 275,000 10s ECG recordings collected in the wild and evaluate it on a range of downstream tasks. The proposed model is a robust backbone for ECG analysis, providing competitive performance on most of the tasks considered, while demonstrating efficacy in low-resource regimes. The code and pre-trained weights are shared publicly at https://github.com/klean2050/tiles_ecg_model., Comment: Pre-print, currently under review

This paper presents the approach and results of USC SAIL's submission to the Signal Processing Grand Challenge 2023 - e-Prevention (Task 2), on detecting relapses in psychotic patients. Relapse prediction has proven to be challenging, primarily due to the heterogeneity of symptoms and responses to treatment between individuals. We address these challenges by investigating the use of sleep behavior features to estimate relapse days as outliers in an unsupervised machine learning setting. We extract informative features from human activity and heart rate data collected in the wild, and evaluate various combinations of feature types and time resolutions. We found that short-time sleep behavior features outperformed their awake counterparts and larger time intervals. Our submission was ranked 3rd in the Task's official leaderboard, demonstrating the potential of such features as an objective and non-invasive predictor of psychotic relapses., Comment: 2 pages, 1 table, ICASSP 2023, Grand Challenges Track

Let $G$ be a connected reductive group acting on a complex vector space $V$ and projective space ${\mathbb P}V$. Let $x\in V$ and ${\cal H}\subseteq {\cal G}$ be the Lie algebra of its stabilizer. Our objective is to understand points $[y]$, and their stabilizers which occur in the vicinity of $[x]$. We construct an explicit ${\cal G}$-action on a suitable neighbourhood of $x$, which we call the local model at $x$. We show that Lie algebras of stabilizers of points in the vicinity of $x$ are parameterized by subspaces of ${\cal H}$. When ${\cal H}$ is reductive these are Lie subalgebras of ${\cal H}$. If the orbit of $x$ is closed this also follows from Luna's theorem. Our construction involves a map connected to the local curvature form at $x$. We apply the local model to forms, when the form $g$ is obtained from the form $f$ as the leading term of a one parameter family acting on $f$. We show that there is a flattening ${\cal K}_0$ of ${\cal K}$, the stabilizer of $f$ which sits as a subalgebra of ${\cal H}$, the stabilizer $g$. We specialize to the case of forms $f$ whose $SL(X)$-orbits are affine, and the orbit of $g$ is of co-dimension $1$. We show that (i) either ${\cal H}$ has a very simple structure, or (ii) conjugates of the elements of ${\cal K}$ also stabilize $g$ and the tangent of exit. Next, we apply this to the adjoint action. We show that for a general matrix $X$, the signatures of nilpotent matrices in its projective orbit closure (under conjugation) are determined by the multiplicity data of the spectrum of $X$. Finally, we formulate the path problem of finding paths with specific properties from $y$ to its limit points $x$ as an optimization problem using local differential geometry. Our study is motivated by Geometric Complexity Theory proposed by the second author and Ketan Mulmuley., Comment: 66 pages

We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as boolean closure of existential fragment of FO via a strengthening of Simon's theorem about piecewise testable languages. We propose a new extension of FO which admits infinitary quantifiers to reason about the inherent infinitary properties of countable words. We provide a very natural and hierarchical block-product based characterization of the new extension. We also explicate its role in view of other natural and classical logical systems such as WMSO and FO[cut] - an extension of FO where quantification over Dedekind-cuts is allowed. We also rule out the possibility of a finite basis for a block-product based characterization of these logical systems. Finally, we report simple but novel algebraic characterizations of one variable fragments of the hierarchies of the new proposed extension of FO.

We develop new algebraic tools to reason about concurrent behaviours modelled as languages of Mazurkiewicz traces and asynchronous automata. These tools reflect the distributed nature of traces and the underlying causality and concurrency between events, and can be said to support true concurrency. They generalize the tools that have been so efficient in understanding, classifying and reasoning about word languages. In particular, we introduce an asynchronous version of the wreath product operation and we describe the trace languages recognized by such products (the so-called asynchronous wreath product principle). We then propose a decomposition result for recognizable trace languages, analogous to the Krohn-Rhodes theorem, and we prove this decomposition result in the special case of acyclic architectures. Finally, we introduce and analyze two distributed automata-theoretic operations. One, the local cascade product, is a direct implementation of the asynchronous wreath product operation. The other, global cascade sequences, although conceptually and operationally similar to the local cascade product, translates to a more complex asynchronous implementation which uses the gossip automaton of Mukund and Sohoni. This leads to interesting applications to the characterization of trace languages definable in first-order logic: they are accepted by a restricted local cascade product of the gossip automaton and 2-state asynchronous reset automata, and also by a global cascade sequence of 2-state asynchronous reset automata. Over distributed alphabets for which the asynchronous Krohn-Rhodes theorem holds, a local cascade product of such automata is sufficient and this, in turn, leads to the identification of a simple temporal logic which is expressively complete for such alphabets.

We develop a new algebraic framework to reason about languages of Mazurkiewicz traces. This framework supports true concurrency and provides a non-trivial generalization of the wreath product operation to the trace setting. A novel local wreath product principle has been established. The new framework is crucially used to propose a decomposition result for recognizable trace languages, which is an analogue of the Krohn-Rhodes theorem. We prove this decomposition result in the special case of acyclic architectures and apply it to extend Kamp's theorem to this setting. We also introduce and analyze distributed automata-theoretic operations called local and global cascade products. Finally, we show that aperiodic trace languages can be characterized using global cascade products of localized and distributed two-state reset automata.

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r-1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component., Comment: New in v4: detailed comparison with Theobald (2009) on page 6. Footnotes removed. Final version accepted by OPERATIONS RESEARCH

This paper extends a recently proposed robust computational framework for constructing the boundary representation (brep) of the volume swept by a given smooth solid moving along a one parameter family $h$ of rigid motions. Our extension allows the input solid to have sharp features, i.e., to be of class G0 wherein, the unit outward normal to the solid may be discontinuous. In the earlier framework, the solid to be swept was restricted to be G1, and thus this is a significant and useful extension of that work. This naturally requires a precise description of the geometry of the surface generated by the sweep of a sharp edge supported by two intersecting smooth faces. We uncover the geometry along with the related issues like parametrization, self-intersection and singularities via a novel mathematical analysis. Correct trimming of such a surface is achieved by a delicate analysis of the interplay between the cone of normals at a sharp point and its trajectory under $h$. The overall topology is explicated by a key lifting theorem which allows us to compute the adjacency relations amongst entities in the swept volume by relating them to corresponding adjacencies in the input solid. Moreover, global issues related to body-check such as orientation are efficiently resolved. Many examples from a pilot implementation illustrate the efficiency and effectiveness of our framework.

This paper proposes a robust algorithmic and computational framework to address the problem of modeling the volume obtained by sweeping a solid along a trajectory of rigid motions. The boundary representation (simply brep) of the input solid naturally induces a brep of the swept volume. We show that it is locally similar to the input brep and this serves as the basis of the framework. All the same, it admits several intricacies: (i) geometric, in terms of parametrizations and, (ii) topological, in terms of orientations. We provide a novel analysis for their resolution. More specifically, we prove a non-trivial lifting theorem which allows to locally orient the output using the orientation of the input. We illustrate the framework by providing many examples from a pilot implementation.

Motivated by multi-distribution divergences, which originate in information theory, we propose a notion of `multi-point' kernels, and study their applications. We study a class of kernels based on Jensen type divergences and show that these can be extended to measure similarity among multiple points. We study tensor flattening methods and develop a multi-point (kernel) spectral clustering (MSC) method. We further emphasize on a special case of the proposed kernels, which is a multi-point extension of the linear (dot-product) kernel and show the existence of cubic time tensor flattening algorithm in this case. Finally, we illustrate the usefulness of our contributions using standard data sets and image segmentation tasks., Comment: To appear in IEEE Computer Society Conference on Computer Vision and Pattern Recognition

We investigate a generalization of the {\L}o\'s-Tarski preservation theorem via the semantic notion of \emph{preservation under substructures modulo $k$-sized cores}. It was shown earlier that over arbitrary structures, this semantic notion for first-order logic corresponds to definability by $\exists^k\forall^*$ sentences. In this paper, we identify two properties of classes of finite structures that ensure the above correspondence. The first is based on well-quasi-ordering under the embedding relation. The second is a logic-based combinatorial property that strictly generalizes the first. We show that starting with classes satisfying any of these properties, the classes obtained by applying operations like disjoint union, cartesian and tensor products, or by forming words and trees over the classes, inherit the same property. As a fallout, we obtain interesting classes of structures over which an effective version of the {\L}o\'s-Tarski theorem holds., Comment: 28 pages, 1 figure

In this work, we propose a detailed computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a "local analysis", which covers parametrization and singularities, as well as a "global theory" which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the "funnel" which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface-surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as either decomposable or non-decomposable. Further, we construct an {\em invariant} function $\theta$ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between $\theta$, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach, $\theta$ is robust and is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included.

We present new preservation theorems that semantically characterize the $\exists^k \forall^*$ and $\forall^k \exists^*$ prefix classes of first order logic, for each natural number $k$. Unlike preservation theorems in the literature that characterize the $\exists^* \forall^*$ and $\forall^* \exists^*$ prefix classes, our theorems relate the count of quantifiers in the leading block of the quantifier prefix to natural quantitative properties of the models. As special cases of our results, we obtain the classical Los-Tarski preservation theorem for sentences in both its extensional and substructural versions. For arbitrary finite vocabularies, we also generalize the extensional version of the Los-Tarski preservation theorem for theories. We also present an interpolant-based approach towards these results. Finally, we present partial results towards generalizing to theories, the substructural version of the Los-Tarski theorem and in the process, we give a preservation theorem that provides a semantic characterization of $\Sigma^0_n$ theories for each natural number $n$., Comment: Added 2 new results: (a) A preservation theorem providing a semantic characterization of \Sigma^0_n theories for each natural number n (which builds on our generalization of the existential amalgamation theorem) (b) Theories in PSC(k) and PSC_f are equivalent to \Sigma^0_2 theories and that the latter are strictly more general than the former. These results are in Sections 8 and 9

We investigate a model-theoretic property that generalizes the classical notion of "preservation under substructures". We call this property \emph{preservation under substructures modulo bounded cores}, and present a syntactic characterization via $\Sigma_2^0$ sentences for properties of arbitrary structures definable by FO sentences. As a sharper characterization, we further show that the count of existential quantifiers in the $\Sigma_2^0$ sentence equals the size of the smallest bounded core. We also present our results on the sharper characterization for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the sharper characterization fails, but for which the classical {\L}o\'s-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the {\L}o\'s-Tarski theorem for some of the aforementioned cases., Comment: From v2 to v3: Corrected typos, edited sentences for better readability; Conjecture 1 of v2 is now resolved so it is now Theorem 4, its proof is included in a new section (Section 7), Thm i in v2 is now Thm i+1 for i >= 4; everything else remains the same. From v1 to v2: Thm i is now Thm i-1 for i >= 7, Corrected the proof of Theorem 10 (now Theorem 9) for B > 2 (statement is still correct)

Sweeping is a powerful and versatile method of designing objects. Boundary of volumes (henceforth envelope) obtained by sweeping solids have been extensively investigated in the past, though, obtaining an accurate parametrization of the envelope remained computationally hard. The present work reports our approach to this problem as well as the important problem of identifying self-intersections within the envelope. Parametrization of the envelope is, of course, necessary for its use in most current CAD systems. We take the more interesting case when the solid is composed of several faces meeting smoothly. We show that the face structure of the envelope mimics locally that of the solid. We adopt the procedural approach at defining the geometry in this work which has the advantage of being accurate as well as computationally efficient. The problem of detecting local self-intersections is central to a robust implementation of the solid sweep. This has been addressed by computing a subtle mathematical invariant which detects self-intersections, and which is computationally benign and requires only point queries.

Given a rank-1 bimatrix game (A,B), i.e., where rank(A+B)=1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in Kannan and Theobald (SODA 2007) and Theobald (2007). In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. This technique also proves the existence, oddness and the index theorem of Nash equilibria in a bimatrix game. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on $[0,1]^k$ for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.

Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the {\em Fisher market game} in which buyers strategize by posing different utility functions. We show that existence of a {\em conflict-free allocation} is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs.

In this paper, we consider the restriction of finite dimensional $GL_{mn} (\C)$-modules to the subgroup ${GL_m (\C)\times GL_n (\C)}$. In particular, for a Weyl module $V_{\lambda} (\C^{mn})$ of $U_q(gl_{mn})$ we construct a representation $W_{\lambda}$ of $U_q (gl_m)\otimes U_q (gl_n)$ such that at $q=1$, the restriction of $V_{\lambda} (\C^{mn})$ to $U_1 (gl_m)\otimes U_1 (gl_n)$ matches its action on $W_{\lambda}$ at $q=1$. Thus $W_{\lambda}$ is a $q$-deformation of the module $V_{\lambda}$. This is achieved by first constructing a $U_q (gl_m)\otimes U_q (gl_n)$-module $\wedge^k $, a $q$-deformation of the simple $GL_{mn} (\C)$-module $\wedge^k (\C^{mn})$. We also construct the bi-crystal basis for $\wedge^k $ and show that it consists of signed subsets. Next, we develop $U_q (gl_m) \otimes U_q (gl_n)$-equivariant maps $\psi_{a,b} :\wedge^{a+1} \otimes \wedge^{b-1} \to \wedge^a \otimes \wedge^b$. This is used as the building block to construct the general $W_{\lambda}$., Comment: 28 pages