@inproceedings{9200,
abstract = {Formal design of embedded and cyber-physical systems relies on mathematical modeling. In this paper, we consider the model class of hybrid automata whose dynamics are defined by affine differential equations. Given a set of time-series data, we present an algorithmic approach to synthesize a hybrid automaton exhibiting behavior that is close to the data, up to a specified precision, and changes in synchrony with the data. A fundamental problem in our synthesis algorithm is to check membership of a time series in a hybrid automaton. Our solution integrates reachability and optimization techniques for affine dynamical systems to obtain both a sufficient and a necessary condition for membership, combined in a refinement framework. The algorithm processes one time series at a time and hence can be interrupted, provide an intermediate result, and be resumed. We report experimental results demonstrating the applicability of our synthesis approach.},
author = {Garcia Soto, Miriam and Henzinger, Thomas A and Schilling, Christian},
booktitle = {HSCC '21: Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control},
isbn = {9781450383394},
keywords = {hybrid automaton, membership, system identification},
location = {Nashville, TN, United States},
pages = {2102.12734},
publisher = {Association for Computing Machinery},
title = {{Synthesis of hybrid automata with affine dynamics from time-series data}},
doi = {10.1145/3447928.3456704},
year = {2021},
}
@inproceedings{9345,
abstract = {Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functionsthat facilitates the efficient search for new materials and material properties. We prove invarianceunder isometries, continuity, and completeness in the generic case, which are necessary featuresfor the reliable comparison of crystals. The proof of continuity integrates methods from discretegeometry and lattice theory, while the proof of generic completeness combines techniques fromgeometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and relatedinclusion-exclusion formulae. We have implemented the algorithm and describe its application tocrystal structure prediction.},
author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin , Vitaliy and Smith, Philip and Wintraecken, Mathijs},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
issn = {1868-8969},
location = {Virtual},
pages = {32:1--32:16},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{The density fingerprint of a periodic point set}},
doi = {10.4230/LIPIcs.SoCG.2021.32},
volume = {189},
year = {2021},
}
@inproceedings{9296,
abstract = { matching is compatible to two or more labeled point sets of size n with labels {1,…,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.},
author = {Aichholzer, Oswin and Arroyo Guevara, Alan M and Masárová, Zuzana and Parada, Irene and Perz, Daniel and Pilz, Alexander and Tkadlec, Josef and Vogtenhuber, Birgit},
booktitle = {15th International Conference on Algorithms and Computation},
isbn = {9783030682101},
issn = {16113349},
location = {Yangon, Myanmar},
pages = {221--233},
publisher = {Springer Nature},
title = {{On compatible matchings}},
doi = {10.1007/978-3-030-68211-8_18},
volume = {12635},
year = {2021},
}
@article{9640,
abstract = {Selection and random drift determine the probability that novel mutations fixate in a population. Population structure is known to affect the dynamics of the evolutionary process. Amplifiers of selection are population structures that increase the fixation probability of beneficial mutants compared to well-mixed populations. Over the past 15 years, extensive research has produced remarkable structures called strong amplifiers which guarantee that every beneficial mutation fixates with high probability. But strong amplification has come at the cost of considerably delaying the fixation event, which can slow down the overall rate of evolution. However, the precise relationship between fixation probability and time has remained elusive. Here we characterize the slowdown effect of strong amplification. First, we prove that all strong amplifiers must delay the fixation event at least to some extent. Second, we construct strong amplifiers that delay the fixation event only marginally as compared to the well-mixed populations. Our results thus establish a tight relationship between fixation probability and time: Strong amplification always comes at a cost of a slowdown, but more than a marginal slowdown is not needed.},
author = {Tkadlec, Josef and Pavlogiannis, Andreas and Chatterjee, Krishnendu and Nowak, Martin A.},
issn = {20411723},
journal = {Nature Communications},
number = {1},
publisher = {Springer Nature},
title = {{Fast and strong amplifiers of natural selection}},
doi = {10.1038/s41467-021-24271-w},
volume = {12},
year = {2021},
}
@phdthesis{9418,
abstract = {Deep learning is best known for its empirical success across a wide range of applications
spanning computer vision, natural language processing and speech. Of equal significance,
though perhaps less known, are its ramifications for learning theory: deep networks have
been observed to perform surprisingly well in the high-capacity regime, aka the overfitting
or underspecified regime. Classically, this regime on the far right of the bias-variance curve
is associated with poor generalisation; however, recent experiments with deep networks
challenge this view.
This thesis is devoted to investigating various aspects of underspecification in deep learning.
First, we argue that deep learning models are underspecified on two levels: a) any given
training dataset can be fit by many different functions, and b) any given function can be
expressed by many different parameter configurations. We refer to the second kind of
underspecification as parameterisation redundancy and we precisely characterise its extent.
Second, we characterise the implicit criteria (the inductive bias) that guide learning in the
underspecified regime. Specifically, we consider a nonlinear but tractable classification
setting, and show that given the choice, neural networks learn classifiers with a large margin.
Third, we consider learning scenarios where the inductive bias is not by itself sufficient to
deal with underspecification. We then study different ways of ‘tightening the specification’: i)
In the setting of representation learning with variational autoencoders, we propose a hand-
crafted regulariser based on mutual information. ii) In the setting of binary classification, we
consider soft-label (real-valued) supervision. We derive a generalisation bound for linear
networks supervised in this way and verify that soft labels facilitate fast learning. Finally, we
explore an application of soft-label supervision to the training of multi-exit models.},
author = {Bui Thi Mai, Phuong},
pages = {125},
publisher = {IST Austria},
title = {{Underspecification in Deep Learning}},
doi = {10.15479/AT:ISTA:9418},
year = {2021},
}
@inproceedings{9543,
abstract = {We consider the problem ofdistributed mean estimation (DME), in which n machines are each given a local d-dimensional vector xv∈Rd, and must cooperate to estimate the mean of their inputs μ=1n∑nv=1xv, while minimizing total communication cost. DME is a fundamental construct in distributed machine learning, and there has been considerable work on variants of this problem, especially in the context of distributed variance reduction for stochastic gradients in parallel SGD. Previous work typically assumes an upper bound on the norm of the input vectors, and achieves an error bound in terms of this norm. However, in many real applications, the input vectors are concentrated around the correct output μ, but μ itself has large norm. In such cases, previous output error bounds perform poorly. In this paper, we show that output error bounds need not depend on input norm. We provide a method of quantization which allows distributed mean estimation to be performed with solution quality dependent only on the distance between inputs, not on input norm, and show an analogous result for distributed variance reduction. The technique is based on a new connection with lattice theory. We also provide lower bounds showing that the communication to error trade-off of our algorithms is asymptotically optimal. As the lattices achieving optimal bounds under l2-norm can be computationally impractical, we also present an extension which leverages easy-to-use cubic lattices, and is loose only up to a logarithmic factor ind. We show experimentally that our method yields practical improvements for common applications, relative to prior approaches.},
author = {Davies, Peter and Gurunanthan, Vijaykrishna and Moshrefi, Niusha and Ashkboos, Saleh and Alistarh, Dan-Adrian},
booktitle = {9th International Conference on Learning Representations},
location = {Virtual},
title = {{New bounds for distributed mean estimation and variance reduction}},
year = {2021},
}
@inproceedings{9416,
abstract = {We study the inductive bias of two-layer ReLU networks trained by gradient flow. We identify a class of easy-to-learn (`orthogonally separable') datasets, and characterise the solution that ReLU networks trained on such datasets converge to. Irrespective of network width, the solution turns out to be a combination of two max-margin classifiers: one corresponding to the positive data subset and one corresponding to the negative data subset. The proof is based on the recently introduced concept of extremal sectors, for which we prove a number of properties in the context of orthogonal separability. In particular, we prove stationarity of activation patterns from some time onwards, which enables a reduction of the ReLU network to an ensemble of linear subnetworks.},
author = {Bui Thi Mai, Phuong and Lampert, Christoph},
booktitle = {9th International Conference on Learning Representations},
location = {Virtual},
title = {{The inductive bias of ReLU networks on orthogonally separable data}},
year = {2021},
}
@article{9225,
abstract = {The Landau–Pekar equations describe the dynamics of a strongly coupled polaron.
Here, we provide a class of initial data for which the associated effective Hamiltonian
has a uniform spectral gap for all times. For such initial data, this allows us to extend the
results on the adiabatic theorem for the Landau–Pekar equations and their derivation
from the Fröhlich model obtained in previous works to larger times.},
author = {Feliciangeli, Dario and Rademacher, Simone Anna Elvira and Seiringer, Robert},
issn = {15730530},
journal = {Letters in Mathematical Physics},
publisher = {Springer Nature},
title = {{Persistence of the spectral gap for the Landau–Pekar equations}},
doi = {10.1007/s11005-020-01350-5},
volume = {111},
year = {2021},
}
@phdthesis{8934,
abstract = {In this thesis, we consider several of the most classical and fundamental problems in static analysis and formal verification, including invariant generation, reachability analysis, termination analysis of probabilistic programs, data-flow analysis, quantitative analysis of Markov chains and Markov decision processes, and the problem of data packing in cache management.
We use techniques from parameterized complexity theory, polyhedral geometry, and real algebraic geometry to significantly improve the state-of-the-art, in terms of both scalability and completeness guarantees, for the mentioned problems. In some cases, our results are the first theoretical improvements for the respective problems in two or three decades.},
author = {Goharshady, Amir Kafshdar},
issn = {2663-337X},
pages = {278},
publisher = {IST Austria},
title = {{Parameterized and algebro-geometric advances in static program analysis}},
doi = {10.15479/AT:ISTA:8934},
year = {2021},
}
@article{7956,
abstract = {When short-range attractions are combined with long-range repulsions in colloidal particle systems, complex microphases can emerge. Here, we study a system of isotropic particles, which can form lamellar structures or a disordered fluid phase when temperature is varied. We show that, at equilibrium, the lamellar structure crystallizes, while out of equilibrium, the system forms a variety of structures at different shear rates and temperatures above melting. The shear-induced ordering is analyzed by means of principal component analysis and artificial neural networks, which are applied to data of reduced dimensionality. Our results reveal the possibility of inducing ordering by shear, potentially providing a feasible route to the fabrication of ordered lamellar structures from isotropic particles.},
author = {Pȩkalski, J. and Rzadkowski, Wojciech and Panagiotopoulos, A. Z.},
issn = {10897690},
journal = {The Journal of chemical physics},
number = {20},
publisher = {AIP},
title = {{Shear-induced ordering in systems with competing interactions: A machine learning study}},
doi = {10.1063/5.0005194},
volume = {152},
year = {2020},
}
@article{7957,
abstract = {Neurodevelopmental disorders (NDDs) are a class of disorders affecting brain development and function and are characterized by wide genetic and clinical variability. In this review, we discuss the multiple factors that influence the clinical presentation of NDDs, with particular attention to gene vulnerability, mutational load, and the two-hit model. Despite the complex architecture of
mutational events associated with NDDs, the various proteins involved appear to converge on common pathways, such as synaptic plasticity/function, chromatin remodelers and the mammalian target of rapamycin (mTOR) pathway. A thorough understanding of the mechanisms behind these pathways will hopefully lead to the identification of candidates that could be targeted for treatment approaches.},
author = {Parenti, Ilaria and Garcia Rabaneda, Luis E and Schön, Hanna and Novarino, Gaia},
issn = {1878108X},
journal = {Trends in Neurosciences},
number = {8},
pages = {608--621},
publisher = {Elsevier},
title = {{Neurodevelopmental disorders: From genetics to functional pathways}},
doi = {10.1016/j.tins.2020.05.004},
volume = {43},
year = {2020},
}
@article{7960,
abstract = {Let A={A1,…,An} be a family of sets in the plane. For 0≤i2b be integers. We prove that if each k-wise or (k+1)-wise intersection of sets from A has at most b path-connected components, which all are open, then fk+1=0 implies fk≤cfk−1 for some positive constant c depending only on b and k. These results also extend to two-dimensional compact surfaces.},
author = {Kalai, Gil and Patakova, Zuzana},
issn = {14320444},
journal = {Discrete and Computational Geometry},
pages = {304--323},
publisher = {Springer Nature},
title = {{Intersection patterns of planar sets}},
doi = {10.1007/s00454-020-00205-z},
volume = {64},
year = {2020},
}
@article{7962,
abstract = {A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.},
author = {Pach, János and Reed, Bruce and Yuditsky, Yelena},
issn = {14320444},
journal = {Discrete and Computational Geometry},
number = {4},
pages = {888--917},
publisher = {Springer Nature},
title = {{Almost all string graphs are intersection graphs of plane convex sets}},
doi = {10.1007/s00454-020-00213-z},
volume = {63},
year = {2020},
}
@inproceedings{7966,
abstract = {For 1≤m≤n, we consider a natural m-out-of-n multi-instance scenario for a public-key encryption (PKE) scheme. An adversary, given n independent instances of PKE, wins if he breaks at least m out of the n instances. In this work, we are interested in the scaling factor of PKE schemes, SF, which measures how well the difficulty of breaking m out of the n instances scales in m. That is, a scaling factor SF=ℓ indicates that breaking m out of n instances is at least ℓ times more difficult than breaking one single instance. A PKE scheme with small scaling factor hence provides an ideal target for mass surveillance. In fact, the Logjam attack (CCS 2015) implicitly exploited, among other things, an almost constant scaling factor of ElGamal over finite fields (with shared group parameters).
For Hashed ElGamal over elliptic curves, we use the generic group model to argue that the scaling factor depends on the scheme's granularity. In low granularity, meaning each public key contains its independent group parameter, the scheme has optimal scaling factor SF=m; In medium and high granularity, meaning all public keys share the same group parameter, the scheme still has a reasonable scaling factor SF=√m. Our findings underline that instantiating ElGamal over elliptic curves should be preferred to finite fields in a multi-instance scenario.
As our main technical contribution, we derive new generic-group lower bounds of Ω(√(mp)) on the difficulty of solving both the m-out-of-n Gap Discrete Logarithm and the m-out-of-n Gap Computational Diffie-Hellman problem over groups of prime order p, extending a recent result by Yun (EUROCRYPT 2015). We establish the lower bound by studying the hardness of a related computational problem which we call the search-by-hypersurface problem.},
author = {Auerbach, Benedikt and Giacon, Federico and Kiltz, Eike},
booktitle = {Advances in Cryptology – EUROCRYPT 2020},
isbn = {9783030457266},
issn = {0302-9743},
pages = {475--506},
publisher = {Springer Nature},
title = {{Everybody’s a target: Scalability in public-key encryption}},
doi = {10.1007/978-3-030-45727-3_16},
volume = {12107},
year = {2020},
}
@article{7968,
abstract = {Organic materials are known to feature long spin-diffusion times, originating in a generally small spin–orbit coupling observed in these systems. From that perspective, chiral molecules acting as efficient spin selectors pose a puzzle that attracted a lot of attention in recent years. Here, we revisit the physical origins of chiral-induced spin selectivity (CISS) and propose a simple analytic minimal model to describe it. The model treats a chiral molecule as an anisotropic wire with molecular dipole moments aligned arbitrarily with respect to the wire’s axes and is therefore quite general. Importantly, it shows that the helical structure of the molecule is not necessary to observe CISS and other chiral nonhelical molecules can also be considered as potential candidates for the CISS effect. We also show that the suggested simple model captures the main characteristics of CISS observed in the experiment, without the need for additional constraints employed in the previous studies. The results pave the way for understanding other related physical phenomena where the CISS effect plays an essential role.},
author = {Ghazaryan, Areg and Paltiel, Yossi and Lemeshko, Mikhail},
issn = {1932-7447},
journal = {The Journal of Physical Chemistry C},
number = {21},
pages = {11716--11721},
publisher = {American Chemical Society},
title = {{Analytic model of chiral-induced spin selectivity}},
doi = {10.1021/acs.jpcc.0c02584},
volume = {124},
year = {2020},
}
@article{7971,
abstract = {Multilayer graphene lattices allow for an additional tunability of the band structure by the strong perpendicular electric field. In particular, the emergence of the new multiple Dirac points in ABA stacked trilayer graphene subject to strong transverse electric fields was proposed theoretically and confirmed experimentally. These new Dirac points dubbed “gullies” emerge from the interplay between strong electric field and trigonal warping. In this work, we first characterize the properties of new emergent Dirac points and show that the electric field can be used to tune the distance between gullies in the momentum space. We demonstrate that the band structure has multiple Lifshitz transitions and higher-order singularity of “monkey saddle” type. Following the characterization of the band structure, we consider the spectrum of Landau levels and structure of their wave functions. In the limit of strong electric fields when gullies are well separated in momentum space, they give rise to triply degenerate Landau levels. In the second part of this work, we investigate how degeneracy between three gully Landau levels is lifted in the presence of interactions. Within the Hartree-Fock approximation we show that the symmetry breaking state interpolates between the fully gully polarized state that breaks C3 symmetry at high displacement field and the gully symmetric state when the electric field is decreased. The discontinuous transition between these two states is driven by enhanced intergully tunneling and exchange. We conclude by outlining specific experimental predictions for the existence of such a symmetry-breaking state.},
author = {Rao, Peng and Serbyn, Maksym},
issn = {2469-9950},
journal = {Physical Review B},
number = {24},
publisher = {American Physical Society},
title = {{Gully quantum Hall ferromagnetism in biased trilayer graphene}},
doi = {10.1103/physrevb.101.245411},
volume = {101},
year = {2020},
}
@article{7985,
abstract = {The goal of limiting global warming to 1.5 °C requires a drastic reduction in CO2 emissions across many sectors of the world economy. Batteries are vital to this endeavor, whether used in electric vehicles, to store renewable electricity, or in aviation. Present lithium-ion technologies are preparing the public for this inevitable change, but their maximum theoretical specific capacity presents a limitation. Their high cost is another concern for commercial viability. Metal–air batteries have the highest theoretical energy density of all possible secondary battery technologies and could yield step changes in energy storage, if their practical difficulties could be overcome. The scope of this review is to provide an objective, comprehensive, and authoritative assessment of the intensive work invested in nonaqueous rechargeable metal–air batteries over the past few years, which identified the key problems and guides directions to solve them. We focus primarily on the challenges and outlook for Li–O2 cells but include Na–O2, K–O2, and Mg–O2 cells for comparison. Our review highlights the interdisciplinary nature of this field that involves a combination of materials chemistry, electrochemistry, computation, microscopy, spectroscopy, and surface science. The mechanisms of O2 reduction and evolution are considered in the light of recent findings, along with developments in positive and negative electrodes, electrolytes, electrocatalysis on surfaces and in solution, and the degradative effect of singlet oxygen, which is typically formed in Li–O2 cells.},
author = {Kwak, WJ and Sharon, D and Xia, C and Kim, H and Johnson, LR and Bruce, PG and Nazar, LF and Sun, YK and Frimer, AA and Noked, M and Freunberger, Stefan Alexander and Aurbach, D},
issn = {0009-2665},
journal = {Chemical Reviews},
number = {14},
pages = {6626--6683},
publisher = {American Chemical Society},
title = {{Lithium-oxygen batteries and related systems: Potential, status, and future}},
doi = {10.1021/acs.chemrev.9b00609},
volume = {120},
year = {2020},
}
@inproceedings{7989,
abstract = {We prove general topological Radon-type theorems for sets in ℝ^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak ε-nets as well as a (p,q)-theorem. More precisely: Let X be either ℝ^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with ℤ₂ coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = ℝ^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface.},
author = {Patakova, Zuzana},
booktitle = {36th International Symposium on Computational Geometry},
isbn = {9783959771436},
issn = {18688969},
location = {Zürich, Switzerland},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Bounding radon number via Betti numbers}},
doi = {10.4230/LIPIcs.SoCG.2020.61},
volume = {164},
year = {2020},
}
@inproceedings{7990,
abstract = {Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on P is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge, removes a non-extreme point of degree 3, or adds a point in P ⧵ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The goal of this paper is to investigate the structure of this graph, with emphasis on its connectivity. For sets P of n points in general position, we show that the bistellar flip graph is (n-3)-connected, thereby answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos) on triangulations. This matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points and projecting the lower convex hull), where (n-3)-connectivity has been known since the late 1980s through the secondary polytope (Gelfand, Kapranov, Zelevinsky) and Balinski’s Theorem. Our methods also yield the following results (see the full version [Wagner and Welzl, 2020]): (i) The bistellar flip graph can be covered by graphs of polytopes of dimension n-3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n-3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations are regular iff the trivial subdivision has height n-3 in the partial order of partial subdivisions. (iv) There are arbitrarily large sets P with non-regular partial triangulations, while every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular partial triangulations (answering a question by F. Santos in the unexpected direction).},
author = {Wagner, Uli and Welzl, Emo},
booktitle = {36th International Symposium on Computational Geometry},
isbn = {9783959771436},
issn = {18688969},
location = {Zürich, Switzerland},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Connectivity of triangulation flip graphs in the plane (Part II: Bistellar flips)}},
doi = {10.4230/LIPIcs.SoCG.2020.67},
volume = {164},
year = {2020},
}
@inproceedings{7991,
abstract = {We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and P is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes the link between "grid peeling" and the ACSF observed by Eppstein et al. (2017), which applied only to convex curves, and which was studied only for regular grids. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.},
author = {Avvakumov, Sergey and Nivasch, Gabriel},
booktitle = {36th International Symposium on Computational Geometry},
isbn = {9783959771436},
issn = {18688969},
location = {Zürich, Switzerland},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Homotopic curve shortening and the affine curve-shortening flow}},
doi = {10.4230/LIPIcs.SoCG.2020.12},
volume = {164},
year = {2020},
}