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Let $G$ be a compact, connected metric graph and let $X\subseteq G$ be a subset. If $X$ is sufficiently dense in $G$, we show that the Gromov--Hausdorff distance matches the Hausdorff distance, namely $d_{GH}(G,X)=d_{H}(G,X)$. In a recent study, when the metric graph is the circle $G=S^1$ with circumference $2\pi$, the equality $d_{GH}(S^1,X)=d_{H}(S^1,X)$ was established whenever $d_{GH}(S^1,X)<\frac{\pi}{6}$. Our result for general metric graphs relaxes this hypothesis in the circle case to $d_{GH}(S^1,X)<\frac{\pi}{3}$, and furthermore, we give an example showing that the constant $\frac{\pi}{3}$ is the best possible. We lower bound the Gromov--Hausdorff distance $d_{GH}(G,X)$ by the Hausdorff distance $d_{H}(G,X)$ via a simple topological obstruction: showing a correspondence with too small distortion contradicts the connectedness of $G$. Moreover, for two sufficiently dense subsets $X,Y\subseteq G$, we provide new lower bounds on $d_{GH}(X,Y)$ in terms of the Hausdorff distance $d_{H}(X,Y)$, which is $O(n^2)$-time computable if the subsets have at most $n$ points.

Modern artificial intelligence (AI) systems are powered by foundation models. This paper presents a new set of foundation models, called Llama 3. It is a herd of language models that natively support multilinguality, coding, reasoning, and tool usage. Our largest model is a dense Transformer with 405B parameters and a context window of up to 128K tokens. This paper presents an extensive empirical evaluation of Llama 3. We find that Llama 3 delivers comparable quality to leading language models such as GPT-4 on a plethora of tasks. We publicly release Llama 3, including pre-trained and post-trained versions of the 405B parameter language model and our Llama Guard 3 model for input and output safety. The paper also presents the results of experiments in which we integrate image, video, and speech capabilities into Llama 3 via a compositional approach. We observe this approach performs competitively with the state-of-the-art on image, video, and speech recognition tasks. The resulting models are not yet being broadly released as they are still under development.

We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let $S^n$ be the $n$-sphere with the geodesic metric, and of diameter $\pi$, and let $\delta > 0$. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex $\mathrm{VR}(S^n;\pi-\delta)$ of the $n$-sphere at scale $\pi-\delta$ occurs in dimension $k$, i.e., suppose that the connectivity is $k-1$. Then $\mathrm{cov}_{S^n}(2k+2) \le \delta < 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$. In other words, there exist $2k+2$ balls of radius $\delta$ that cover $S^n$, and no set of $k$ balls of radius $\frac{\delta}{2}$ cover the projective space $\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$ changes infinitely many times as the scale $r$ increases.

Electromagnetic stimulation probes and modulates the neural systems that control movement. Key to understanding their effects is the muscle recruitment curve, which maps evoked potential size against stimulation intensity. Current methods to estimate curve parameters require large samples; however, obtaining these is often impractical due to experimental constraints. Here, we present a hierarchical Bayesian framework that accounts for small samples, handles outliers, simulates high-fidelity data, and returns a posterior distribution over curve parameters that quantify estimation uncertainty. It uses a rectified-logistic function that estimates motor threshold and outperforms conventionally used sigmoidal alternatives in predictive performance, as demonstrated through cross-validation. In simulations, our method outperforms non-hierarchical models by reducing threshold estimation error on sparse data and requires fewer participants to detect shifts in threshold compared to frequentist testing. We present two common use cases involving electrical and electromagnetic stimulation data and provide an open-source library for Python, called hbMEP, for diverse applications.

We prove that for each positive integer $n$, the Rips complexes of the $n$-dimensional integer lattice in the $d_1$ metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales above $n^2(2n-1)$, with the bounds arising from the Jung's constants. We introduce a new concept of locally dominated vertices in a simplicial complex, upon which our proof strategy is based. This allows us to deduce the contractibility of the Rips complexes from a local geometric condition called local crushing. In the case of the integer lattices in dimension $n$ and a fixed scale $r$, this condition entails the comparison of finitely many distances to conclude that the corresponding Rips complex is contractible. In particular, we are able to verify that for $n=1,2,3$, the Rips complex of the $n$-dimensional integer lattice at scale greater or equal to $n$ is contractible. We conjecture that the same proof strategy can be used to extend this result to all dimensions $n$, Comment: 18 pages, 3 figures

A good summary can often be very useful during program comprehension. While a brief, fluent, and relevant summary can be helpful, it does require significant human effort to produce. Often, good summaries are unavailable in software projects, thus making maintenance more difficult. There has been a considerable body of research into automated AI-based methods, using Large Language models (LLMs), to generate summaries of code; there also has been quite a bit work on ways to measure the performance of such summarization methods, with special attention paid to how closely these AI-generated summaries resemble a summary a human might have produced. Measures such as BERTScore and BLEU have been suggested and evaluated with human-subject studies. However, LLM-produced summaries can be too long, irrelevant, etc: generally, too dissimilar to what a human might say. Given an LLM-produced code summary, how can we judge if a summary is good enough? Given some input source code, and an LLM-generated summary, existing approaches can help judge brevity, fluency and relevance; however, it's difficult to gauge whether an LLM-produced summary sufficiently resembles what a human might produce, without a "golden" human-produced summary to compare against. We study this resemblance question as a calibration problem: given just the summary from an LLM, can we compute a confidence measure, that provides a reliable indication of whether the summary sufficiently resembles what a human would have produced in this situation? We examine this question using several LLMs, for several languages, and in several different settings. Our investigation suggests approaches to provide reliable predictions of the likelihood that an LLM-generated summary would sufficiently resemble a summary a human might write for the same code.

Lexical Semantic Change Detection (LSCD) is a complex, lemma-level task, which is usually operationalized based on two subsequently applied usage-level tasks: First, Word-in-Context (WiC) labels are derived for pairs of usages. Then, these labels are represented in a graph on which Word Sense Induction (WSI) is applied to derive sense clusters. Finally, LSCD labels are derived by comparing sense clusters over time. This modularity is reflected in most LSCD datasets and models. It also leads to a large heterogeneity in modeling options and task definitions, which is exacerbated by a variety of dataset versions, preprocessing options and evaluation metrics. This heterogeneity makes it difficult to evaluate models under comparable conditions, to choose optimal model combinations or to reproduce results. Hence, we provide a benchmark repository standardizing LSCD evaluation. Through transparent implementation results become easily reproducible and by standardization different components can be freely combined. The repository reflects the task's modularity by allowing model evaluation for WiC, WSI and LSCD. This allows for careful evaluation of increasingly complex model components providing new ways of model optimization.

Let $n$ be a positive integer. We provide an explicit geometrically motivated $1$-Lipschitz map from the space of persistence diagrams on $n$ points (equipped with the Bottleneck distance) into Hilbert space. Such maps are a crucial step in topological data analysis, allowing the use of statistic (and thus data analysis) on collections of persistence diagrams. The main advantage of our maps as compared to most of the other such transformations is that they are coarse and uniform embeddings with explicit distortion functions. This allows us to control the amount of geometric information lost through their application. Furthermore, we provide an explicit $1$-Lipschitz map from the space of persistence diagrams on $n$ points on a bounded domain into a Euclidean space with an explicit distortion function. The mentioned maps are fairly simple, with each component depending depending only on the bottleneck distance to the corresponding landmark persistence diagram. Due to geometric motivation from classical dimension theory, our methods are best described as quantitative dimension theory. We discuss the advantages and disadvantages of our approach. We conclude with differently flavoured embedding of the space of persistence diagrams on $n$ points on a bounded domain into $\mathbb{R}^{n(n+1)}$., Comment: 18 pages

We present the DURel tool that implements the annotation of semantic proximity between uses of words into an online, open source interface. The tool supports standardized human annotation as well as computational annotation, building on recent advances with Word-in-Context models. Annotator judgments are clustered with automatic graph clustering techniques and visualized for analysis. This allows to measure word senses with simple and intuitive micro-task judgments between use pairs, requiring minimal preparation efforts. The tool offers additional functionalities to compare the agreement between annotators to guarantee the inter-subjectivity of the obtained judgments and to calculate summary statistics giving insights into sense frequency distributions, semantic variation or changes of senses over time., Comment: EACL Demo, 7 pages

Lattices and periodic point sets are well known objects from discrete geometry. They are also used in crystallography as one of the models of atomic structure of periodic crystals. In this paper we study the embedding properties of spaces of lattices and periodic point sets equipped with optimal bijection metrics (i.e., bottleneck and Euclidean bottleneck metrics). We focus our treatment on embeddings into Hilbert space. On one hand this is motivated by modern data analysis, which is mostly based on statistical approaches developed on Euclidean on Hilbert spaces, hence such embeddings play a major role in applied pipelines. On the other hand there is a well-established methodology related to such questions in coarse geometry, arising from the work on the Novikov conjecture. The main results of this paper provide different conditions, under which the spaces of lattices or periodic point sets are Lipschitz or coarsely (non)embeddable into Hilbert space. The various conditions are phrased in terms of density, packing radius, covering radius, the cardinality of the motif, and the diameter of the unit cell.

We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let $VR(Q_n;r)$ be its Vietoris--Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r

We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X,d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is induced by local minima of the distance function $d$. When $d$ attains local minimum at only finitely many pairs of points, we prove that each above mentioned change in persistence is induced by a specific critical edge in Rips complexes, which represents a local minimum of $d$. We use this fact to develop a theory (including interpretation) of critical edges of persistence. The obtained results include upper bounds for the rank of one-dimensional persistence and a corresponding reconstruction result. Of potential computational interest is a simple geometric criterion recognizing local minima of $d$ that induce a change in persistence. We conclude with a proof that each locally isolated minimum of $d$ can be detected through persistent homology with selective Rips complexes. The results of this paper offer the first interpretation of critical scales of persistent homology (obtained via Rips complexes) for general compact metric spaces., Comment: 20 pages, 5 figures

Understanding the landscape of potential harms from algorithmic systems enables practitioners to better anticipate consequences of the systems they build. It also supports the prospect of incorporating controls to help minimize harms that emerge from the interplay of technologies and social and cultural dynamics. A growing body of scholarship has identified a wide range of harms across different algorithmic technologies. However, computing research and practitioners lack a high level and synthesized overview of harms from algorithmic systems. Based on a scoping review of computing research $(n=172)$, we present an applied taxonomy of sociotechnical harms to support a more systematic surfacing of potential harms in algorithmic systems. The final taxonomy builds on and refers to existing taxonomies, classifications, and terminologies. Five major themes related to sociotechnical harms - representational, allocative, quality-of-service, interpersonal harms, and social system/societal harms - and sub-themes are presented along with a description of these categories. We conclude with a discussion of challenges and opportunities for future research.

Some hard sphere colloidal models have been criticized for inaccurately predicting the solution viscosity of complex biological molecules like proteins. Competing short-range attractions and long-range repulsions, also known as SALR interactions, have been thought to affect the microstructure of a protein solution at low to moderate ionic strength. However, such interactions have been implicated primarily in causing phase transition, protein gelation, or reversible cluster formation and their effect on protein solution viscosity change is not fully understood. In this work we show the application of a hard sphere colloidal model with SALR interactions towards predicting the viscosity of dilute to semi-dilute protein solutions. The comparison is performed for a globular shaped albumin and Y-shaped therapeutic monoclonal antibody that are not explained by previous colloidal models. The model predictions show that it is the coupling between attractions and repulsions that give rise to the observed experimental trends in solution viscosity as a function of pH, concentration, and ionic strength. The parameters of the model are obtained from measurements of the second virial coefficient and net surface charge/zeta-potential, without additional fitting of the viscosity.

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris complex contains all simplices with vertex set contained in some $U \in \mathcal{U}$, and the Vietoris metric thickening is the space of probability measures with support in some $U \in \mathcal{U}$, equipped with an optimal transport metric. We show that the Vietoris metric thickening and the Vietoris complex have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover $\mathcal{U}$ appropriately, we get isomorphisms between the homotopy groups of Vietoris--Rips metric thickenings and simplicial complexes, where both spaces are defined using the convention ``diameter $< r$'' (instead of $\le r$). Similarly, we get isomorphisms between the homotopy groups of \v{C}ech metric thickenings and simplicial complexes, where both spaces are defined using open balls (instead of closed balls).

Selective Rips complexes corresponding to a sequence of parameters are a generalization of Vietoris-Rips complexes utilizing the idea of thin simplices. We prove that if a metric space $Y$ is close (in Gromov-Hausdorff distance) to a closed Riemannian manifold $X$, then selective Rips complexes of $Y$ for certain parameters attain the homotopy type of $X$. This result is a generalization of Latchev's reconstruction result from Vietoris-Rips complexes to selective Rips complexes. In particular, we present a novel proof for the Latschev's theorem as a special case. We also present a functorial setting, which is new even in the case of Vietoris-Rips complexes., Comment: 19 pages, 3 figure

Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the critical simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration functions. In this paper we provide a sufficient condition for rigidity of a terminal simplex, i.e., a condition on $\epsilon > 0$ implying that the terminal simplex of a homology class or a bar in persistent homology remains constant through $\epsilon$-perturbations of filtration function. The condition for a homology class or a bar in dimension n depends on the barcodes in dimensions n and n+1., Comment: 21 pages, 6 figures

Federated Learning (FL) enables multiple parties to distributively train a ML model without revealing their private datasets. However, it assumes trust in the centralized aggregator which stores and aggregates model updates. This makes it prone to gradient tampering and privacy leakage by a malicious aggregator. Malicious parties can also introduce backdoors into the joint model by poisoning the training data or model gradients. To address these issues, we present BEAS, the first blockchain-based framework for N-party FL that provides strict privacy guarantees of training data using gradient pruning (showing improved differential privacy compared to existing noise and clipping based techniques). Anomaly detection protocols are used to minimize the risk of data-poisoning attacks, along with gradient pruning that is further used to limit the efficacy of model-poisoning attacks. We also define a novel protocol to prevent premature convergence in heterogeneous learning environments. We perform extensive experiments on multiple datasets with promising results: BEAS successfully prevents privacy leakage from dataset reconstruction attacks, and minimizes the efficacy of poisoning attacks. Moreover, it achieves an accuracy similar to centralized frameworks, and its communication and computation overheads scale linearly with the number of participants., Comment: The Third AAAI Workshop on Privacy-Preserving Artificial Intelligence (PPAI-22) at the Thirty-Sixth AAAI Conference on Artificial Intelligence (AAAI-22)

We prove that the existence of a $1$-Lipschitz retraction (a contraction) from a space $X$ onto its subspace $A$ implies the persistence diagram of $A$ embeds into the persistence diagram of $X$. As a tool we introduce tight injections of persistence modules as maps inducing the said embeddings. We show contractions always exist onto shortest loops in metric graphs and conjecture on existence of contractions in planar metric graphs onto all loops of a shortest homology basis. Of primary interest are contractions onto loops in geodesic spaces. These act as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and Adams describes the pattern of persistence diagram of $S^1$, a contraction $X \to S^1$ implies the same pattern appears in persistence diagram of $X$., Comment: 12 pages, 3 figures

Federated learning enables multiple data owners to jointly train a machine learning model without revealing their private datasets. However, a malicious aggregation server might use the model parameters to derive sensitive information about the training dataset used. To address such leakage, differential privacy and cryptographic techniques have been investigated in prior work, but these often result in large communication overheads or impact model performance. To mitigate this centralization of power, we propose SCOTCH, a decentralized m-party secure-computation framework for federated aggregation that deploys MPC primitives, such as secret sharing. Our protocol is simple, efficient, and provides strict privacy guarantees against curious aggregators or colluding data-owners with minimal communication overheads compared to other existing state-of-the-art privacy-preserving federated learning frameworks. We evaluate our framework by performing extensive experiments on multiple datasets with promising results. SCOTCH can train the standard MLP NN with the training dataset split amongst 3 participating users and 3 aggregating servers with 96.57% accuracy on MNIST, and 98.40% accuracy on the Extended MNIST (digits) dataset, while providing various optimizations., Comment: Thirty-Sixth AAAI Conference on Artificial Intelligence (AAAI-22), Third AAAI Privacy-Preserving Artificial Intelligence (PPAI-22) Workshop

A discussion of homotopy limits of (1-)stacks, with an emphasis on fixed point stacks., Comment: Exposition streamlined. Minor errors corrected. Discussion of derived functors added

This note records that in the setting of complex varieties, the cohomological consequence of Ehresmann's fibration theorem holds without the smooth assumption on the base or the total space.

This note records that the Langlands parameter spaces, associated by Adams- Barbasch-Vogan to a real group, may be described as homotopy fixed points (fixed point stacks) of the spaces associated to the corresponding complex group., Comment: Minor grammatical fixes

In this work, we analyzed an isotropic colloidal model incorporating both short-range sticky attractions and long-range electrostatic repulsions. We computed the zero-shear viscosity and second virial coefficient for a dilute colloidal suspension (i.e., pair interactions only) as a function of the strength of attractions and repulsions. We also developed an analytical approximation that allows us to better understand the coupling of the two types of interactions. The attractions and repulsions contribute to the zero-shear viscosity and second virial coefficient in different ways, leading to cases with the same second virial coefficient but different zero-shear viscosity. The analytical approximation shows that the mechanism of the coupling of interactions is that long-range repulsions can weaken the influence of short-range attractions. This effect alters how repulsions change the zero-shear viscosity. Acting independently, both attractions and repulsions increase the viscosity coefficient of the system. However, when both type of interactions are considered together, repulsions can screen the effect of attractive interactions thereby reducing the viscosity.

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions we show that such a loop either terminates a $1$-dimensional homology class or gives rise to a $2$-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology in order to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes., Comment: 19 pages, 9 figures

Medical imaging datasets are inherently high dimensional with large variability and low sample sizes that limit the effectiveness of deep learning algorithms. Recently, generative adversarial networks (GANs) with the ability to synthesize realist images have shown great potential as an alternative to standard data augmentation techniques. Our work focuses on cross-modality synthesis of fluorodeoxyglucose~(FDG) Positron Emission Tomography~(PET) scans from structural Magnetic Resonance~(MR) images using generative models to facilitate multi-modal diagnosis of Alzheimer's disease (AD). Specifically, we propose a novel end-to-end, globally and locally aware image-to-image translation GAN (GLA-GAN) with a multi-path architecture that enforces both global structural integrity and fidelity to local details. We further supplement the standard adversarial loss with voxel-level intensity, multi-scale structural similarity (MS-SSIM) and region-of-interest (ROI) based loss components that reduce reconstruction error, enforce structural consistency at different scales and perceive variation in regional sensitivity to AD respectively. Experimental results demonstrate that our GLA-GAN not only generates synthesized FDG-PET scans with enhanced image quality but also superior clinical utility in improving AD diagnosis compared to state-of-the-art models. Finally, we attempt to interpret some of the internal units of the GAN that are closely related to this specific cross-modality generation task., Comment: 24 pages

Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are geometric manifestations of the corresponding holes and often carry desirable information. We propose a new method of extraction of persistent homology representatives using cohomology. In a nutshell, we first compute persistent cohomology and use the obtained information to significantly improve the running time of the direct persistent homology computations. This algorithm applied to Rips filtrations generally computes persistent homology representatives much faster than the standard methods., Comment: 16 pages, 4 figures, accepted for publication in Accepted for publication in Foundations of Data Science

Scattering by an isolated defect embedded in a dielectric medium of two dimensional periodicity is of interest in many sub-fields of electrodynamics. Present approaches to compute this scattering rely either on the Born approximation and its quasi-analytic extensions, or on \emph{ab-initio} computation that requires large domain sizes to reduce the effects of boundary conditions. The Born approximation and its extensions are limited in scope, while the ab-initio approach suffers from its high numerical cost. In this paper, I introduce a hybrid scheme in which an effective local electric susceptibility tensor of a defect is estimated by solving an inverse problem efficiently. The estimated tensor is embedded into an S-matrix formula based on the reciprocity theorem. With this embedding, the computation of the S-matrix of the defect requires field solutions only in the unit cell of the background. In practice, this scheme reduces the computational cost by almost two orders of magnitude, while sacrificing little in accuracy. The scheme demonstrates that statistical estimation can capture sufficient information from cheap calculations to compute quantities in the far field. I outline the fundamental theory and algorithms to carry out the computations in high dielectric contrast materials, including metals. I demonstrate the capabilities of this approach with examples from optical inspection of nano-electronic circuitry where the Born approximation fails and the existing methods for its extension are also inapplicable.

Given a metric space $X$ and a subspace $A\subset X$, we prove $A$ can generate various algebraic elements in persistent homology of $X$. We call such elements (algebraic) footprints of $A$. Our results imply that footprints typically appear in dimensions above the dimension of $A$. Higher-dimensional persistent homology thus encodes lower-dimensional geometric features of $X$. We pay special attention to a specific type of geodesics in a geodesic surface $X$ called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and length spectrum in Riemannian geometry., Comment: 26 pages, 9 figures; Accepted for publication in Mediterranean Journal of Mathematics

Vector control strategies are central to the mitigation and containment of COVID-19 and have come in the form of municipal ordinances that restrict the operational status of public and private spaces and associated services. Yet, little is known about specific population responses in terms of risk behaviors. To help understand the impact of those vector control variable strategies, a multi-week, multi-site observational study was undertaken outside of 19 New York City medical facilities during the peak of the city's initial COVID-19 wave (03/22/20-05/19/20). The aim was to capture perishable data of the touch, destination choice, and PPE usage behavior of individuals egressing hospitals and urgent care centers. A major goal was to establish an empirical basis for future research on the way people interact with three-dimensional vector environments. Anonymized data were collected via smart phones. Each data record includes the time, data, and location of an individual leaving a healthcare facility, their routing, interactions with the build environment, other individuals, and themselves. Most records also note their PPE usage, destination, intermediary stops, and transportation choices. The records were linked with 61 socio-economic factors by the facility zip code and 7 contemporaneous weather factors and the merged in a unified shapefile in an ARCGIS system. This paper describes the project team and protocols used to produce over 5,100 publicly accessible observational records and an affiliated codebook that can be used to study linkages between individual behaviors and on-the-ground conditions., Comment: 14 Pages, 4 figures

Limited availability of annotated medical imaging data poses a challenge for deep learning algorithms. Although transfer learning minimizes this hurdle in general, knowledge transfer across disparate domains is shown to be less effective. On the other hand, smaller architectures were found to be more compelling in learning better features. Consequently, we propose a lightweight architecture that uses mixed asymmetric kernels (MAKNet) to reduce the number of parameters significantly. Additionally, we train the proposed architecture using semi-supervised learning to provide pseudo-labels for a large medical dataset to assist with transfer learning. The proposed MAKNet provides better classification performance with $60 - 70\%$ less parameters than popular architectures. Experimental results also highlight the importance of domain-specific knowledge for effective transfer learning., Comment: 9 pages 4 figures

Numerical simulations are performed on a stochastic model based on Monte Carlo damage simulation process and Markov Chain Monte Carlo techniques to investigate the formation and evaluation of isolated and multiple DNA damage and cellular survival by light ionizing radiation in a colony of tumour cells. The contribution of the local clustering of the strand breaks and base damage is taken into account while considering double-strand breaks (DSBs) as primary lesions in the DNA of the cell nucleus induced by ionizing radiation. The model incorporates the combined effects of biological processes such as the tumour oxygenation, cellular multiplication and mutation through various probability distributions in a full Monte Carlo simulation of fractionated hadrontherapy. Our results indicate that the linear and quadratic parameters of the model show a negative correlation, for protons and helium ions, which might suggest an underlying biological mechanism. Despite using a model with quite different descriptions of linear and quadratic parameters, the observed results for linear parameter show largely reasonable agreement while the quadratic parameter consistent deviations from the results obtained using the LEM model at low LET. In addition to the LET dependence, RBE values showed a strong dependence on alpha/beta ratio and a considerable scatter for various particle types indicate particle specific behaviour of initial its slope. The surviving curves show a non-linear dose-response suggesting that interaction among DSBs induced by ionizing radiation contribute significantly to the quadratic term of the model., Comment: 17 pages and 8 figures

Using ideas of the Dowker duality we prove that the Rips complex at scale $r$ is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. These results are incorporated into a systematic theory of filtrations arising from covers. As a result we provide a general framework for reconstruction of spaces by Rips complexes, a short proof of the reconstruction result of Hausmann, and completely classify reconstruction scales for metric graphs. Furthermore we introduce a new extraction method for homology of a space based on nested Rips complexes at a single scale, which requires no conditions on neighboring scales nor the Euclidean structure of the ambient space., Comment: 19 pages, some issues resolved

In this paper, we present iPrescribe, a scalable low-latency architecture for recommending 'next-best-offers' in an online setting. The paper presents the design of iPrescribe and compares its performance for implementations using different real-time streaming technology stacks. iPrescribe uses an ensemble of deep learning and machine learning algorithms for prediction. We describe the scalable real-time streaming technology stack and optimized machine-learning implementations to achieve a 90th percentile recommendation latency of 38 milliseconds. Optimizations include a novel mechanism to deploy recurrent Long Short Term Memory (LSTM) deep learning networks efficiently., Comment: 7 Pages, Accepted in COMAD-CODS 2019

We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when the number of points is not bounded, the corresponding spaces of persistence diagrams do not have finite asymptotic dimension. Furthermore, in the case of the bottleneck distance, the corresponding space does not coarsely embed into Hilbert space., Comment: Error in a proof corrected, see the note in the introduction for details

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable usage of existing computational topology software in this context.

We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic approximations of it. The definitions turn out to be equivalent and give rise to a full six-functor formalism. For rational \'etale motives over a finite field or the homotopical stable algebraic derivator arising from the semisimplified Hodge realization, the equivariant mixed Tate motives provide a graded version of the equivariant derived category. We show that, in sufficiently nice and clean cases, these categories admit weight structures; moreover, a tilting result holds which identifies the category of equivariant mixed Tate motives with the bounded homotopy category of the heart of its weight structure. This can be seen as a formality result for equivariant derived categories. We also discuss convolution functors on equivariant mixed Tate motives, and consequences for the categorification of the Hecke algebra and some of its modules., Comment: 198 pages, includes an appendix on tilting results by F. H\"ormann and M. Wendt. Comments welcome!

We theoretically study the electronic structure and spin properties of one-dimensional nanostructures of the prototypical bulk topological insulator Bi$_2$Se$_3$. Realistic models of experimentally observed Bi$_2$Se$_3$ nanowires and nanoribbons are considered using the tight-binding method. At low energies, the band structures are composed of a series of evenly spaced degenerate sub-bands resulting from circumferential confinement of the topological surface states. The direct band gaps due to the non-trivial $\pi$ Berry phase show a clear dependence on the circumference. The spin-momentum locking of the topological surface states results in a pronounced 2$\pi$ spin rotation around the circumference with the degree of spin polarization dependent on the the momentum along the nanostructure. Overall, the band structures and spin textures are more complicated for nanoribbons, which expose two distinct facets. The effects of reduced dimensionality are rationalized with the help of a simple model that considers circumferential quantization of the topological surface states. Furthermore, the surface spin density induced by electric current along the nanostructure shows a pronounced oscillatory dependence on the charge-carrier energy, which can be exploited in spintronics applications., Comment: 10 pages, 9 figures

A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space $X$ carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of $X$ may be obtained by an appropriate finite sample (subset of $X$), and that persistence of any subset of $X$ is well interleaved with the persistence of $X$. It follows that the persistence of $X$ is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each $r>0$ a density $s>0$, so that for each $s$-dense sample $S \subset X$ the corresponding fundamental group (and the first homology) of the Rips complex of $S$ is isomorphic to the one of $X$, leading to an improved reconstruction result., Comment: 18 pages, 7 figures, a bit reorganised in the last version

Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or \v{C}ech filtration of $X$ to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and \v{C}ech induced persistences. Amongst other results we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base and that Rips and \v{C}ech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence., Comment: 35 pages, 8 figures. Definition 4.1 has been updated. Now it also holds for non semi-locally simply connected spaces. The details of this modification are in the last section "Comments on this version of the manuscript"

A variation on the splitting principle

Let $X$ and $Y$ be proper metric spaces. We show that a coarsely $n$-to-$1$ map $f\colon X\to Y$ induces an $n$-to-$1$ map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if $f \colon X\to Y$ is a coarsely $n$-to-$1$ map between proper metric spaces $X$ and $Y$ then $asdim(Y) \leq asdim(X) + n -1$. Furthermore we introduce coarsely open coarsely $n$-to-$1$ maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension., Comment: An updated version (in 2021) containing a small correction, for details see Acknowledgments

Binary bismuth chalcogenides Bi$_2$Se$_3$, Bi$_2$Te$_3$, and related materials are currently being extensively investigated as the reference topological insulators (TIs) due to their simple surface-state band dispersion (single Dirac cone) and relatively large bulk band gaps. Nanostructures of TIs are of particular interest as an increased surface-to-volume ratio enhances the contribution of surfaces states, meaning they are promising candidates for potential device applications. So far, the vast majority of research efforts have focused on the low-energy (0001) surfaces, which correspond to natural cleavage planes in these layered materials. However, the surfaces of low-dimensional nanostructures (nanoplatelets, nanowires, nanoribbons) inevitably involve higher-index facets. We perform a systematic ab initio investigation of the surfaces of bismuth chalcogenide TI nanostructures characterized by different crystallographic orientations, atomic structures and stoichiometric compositions. We find several stable terminations of high-index surfaces, which can be realized at different values of the chemical potential of one of constituent elements. For the uniquely defined stoichiometric termination, the topological Dirac fermion states are shown to be strongly anisotropic with a clear dependence of Fermi velocities and spin polarization on the surface orientation. Self-doping effects and the presence of topologically trivial mid-gap states are found to characterize the non-stoichiometric surfaces. The results of our study pave the way towards experimental control of topologically protected surface states in bismuth chalcogenide nanostructures.

We study the perturbation by a critical term and a $(p-1)$-superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem.

Observations on rational Chow groups and cycle class maps in equivariant contexts., Comment: Exposition streamlined; references added

A simple explanation for the ubiquity of "vanishing of cohomology in odd degrees" in equivariant contexts is given., Comment: References plus some complements added

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the class of visual functions $f$ that do induce continuous maps $\tilde f$ between Gromov boundaries. Its subclass, the class of radial functions, induces Hoelder maps between Gromov boundaries. Conversely, every Hoelder map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large scale properties of f and small scale properties of $f$, especially related to the dimension theory. In particular, we prove a form of the dimension raising theorem. We give a natural example of a radial dimension raising map and we also give a general class of radial functions that raise asymptotic dimension.

The aim of this paper is to investigate properties preserved and co-preserved by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to X/\sim$ induced by a finite group $G$ acting by isometries on a metric space $X$. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C, straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.

Hydrogen adatoms and other species covalently bound to graphene act as resonant scattering centers affecting the electronic transport properties and inducing Anderson localization. We show that attractive interactions between adatoms on graphene and their diffusion mobility strongly modify the spatial distribution, thus fully eliminating isolated adatoms and increasing the population of larger size adatom aggregates. Our scaling analysis shows that such aggregation of adatoms increases conductance by up to several orders of magnitude and results in significant extension of the Anderson localization length in the strong localization regime. We introduce a simple definition of the effective adatom concentration $x^\star$ , which describes the transport properties of both random and correlated distributions of hydrogen adatoms on graphene across a broad range of concentrations., Comment: Main text: 5 pages, 5 figures. Supplementary material: 7 pages, 5 figures

It is well known that the Euler characteristic of the cohomology of a complex algebraic variety coincides with the Euler characteristic of its cohomology with compact support. An old result of G. Laumon asserts that a relative version of this statement is true in $\ell$-adic cohomology. The purpose of this note is to extend Laumon's result to the topological setting. Some applications are also discussed., Comment: proof of Theorem 4.1 clarified