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Francisco Santos

Publications and source records attributed to Francisco Santos.

At least 19 recordsLinked to original sources

Adversarial Attacks on Downstream Weather Forecasting Models: Application to Tropical Cyclone Trajectory Prediction

Deep learning-based weather forecasting (DLWF) models leverage past weather observations to generate future forecasts, supporting a wide range of downstream applications, including tropical cyclone (TC) prediction. In this paper, we investigate their vulnerability to adversarial attacks, where subtle perturbations to the upstream forecasts can alter the downstream TC trajectory predictions. Although research into adversarial attacks on DLWF models has grown recently, it remains challenging to craft perturbed upstream forecasts that steer the downstream outputs toward attacker-specified trajectories. First, conventional TC detection systems are opaque, non-differentiable black boxes, making standard gradient-based attacks infeasible. Second, the extreme rarity of TC events leads to severe class imbalance problem, making it difficult to develop attack methods for perturbing upstream forecasts that produce realistic-looking cyclone paths aligned with attacker's target trajectories. To overcome these limitations, we propose Cyc-Attack, a novel method for perturbing the upstream forecasts of DLWF models to generate adversarial trajectories. The proposed method uses a differentiable surrogate model to approximate the TC detector's output, enabling the application of gradient-based attacks. Cyc-Attack also employs a skewness-aware loss function with kernel dilation strategy to address the imbalance problem. Finally, a distance-based gradient weighting scheme and regularization are used to constrain the perturbations and eliminate unrealistic-looking trajectories, thereby making the adversarial upstream forecasts less easily detectable. Our experiments show that Cyc-Attack achieves a higher true positive rate in matching the attacker's target trajectories, along with lower false alarm rates and stealthier perturbations than conventional attack methods.

cs.LG

Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture

Henze and Malikiosis (2017) have shown that the Lonely Runner Conjecture (LRC) can be restated as a convex-geometric question on the so-called LR zonotopes, lattice zonotopes with one more generator than their dimension. This relation naturally suggests a more generel statement, the "shifted" LRC, the zonotopal version of which concerns a classical parameter, the covering radius. Theorems A and B in Malikiosis-Schymura-Santos (2025) use the zonotopal restatements of both the original and the shifted LRC to prove a linearly-exponential bound on the size of the (integer) speeds for which the conjectures need to be checked in order to establish them for each fixed number of runners; in the shifted version their statement and proof rely on a certain assumption on two-dimensional rational vector configurations, the so-called "Lonely Vector Property". In this paper we do two things: We push the analogies between the two versions of LRC and their zonotopal counterparts, in particular highlighting that the proofs of Theorems A and B in Malikiosis-Schymura-Santos are more transparent, and the statments more general, if regarded in terms of two quite general classes of lattice zonotopes: the coloopless zonotopes that we introduce here and the cosimple ones, already defined by them. These classes contain all primitive zonotopes of widths at least two and at least three, respectively. We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at $n=5$) and to the Lonely Vector Property (starting at $n=12$).

math.CO

Moduli spaces of Delzant polytopes and symplectic toric manifolds

This paper introduces modern geometric combinatorial technology from the theory of triangulations in order to derive results in toric symplectic geometry. In the main part of the paper we prove a number of properties of the space $\mathcal{D}(n)$ of $n$-dimensional Delzant polytopes. Two highlights are the construction of examples showing that, in contrast with the classical work of Oda in dimension $2$, no classification of combinatorially minimal Delzant polytopes can be expected in dimension $3$ or higher, and a proof that the space of $n$-dimensional Delzant polytopes is path-connected. Our proof of the latter is based on the fact that every rational fan can be refined to a unimodular fan, which is a standard technique used for resolution of singularities of toric varieties. In the last part of the paper, using the Delzant correspondence, these results allow us to answer several open questions concerning the moduli space $\mathcal{M}(n)$ of symplectic toric manifolds of dimension $2n$, since this space is isometric to the space of Delzant polytopes. Our results imply that no classification of minimal models of symplectic toric manifolds is plausible in dimension $6$ or higher, which answers in the negative a long-standing folklore question originating in Oda's work (1978).

math.SG

Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to $n+1$ runners it suffices to consider positive integer velocities in the order of $n^{O(n^2)}$. Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to $\binom{n+1}{2}^{n-1} \le n^{2n}$ are enough. We prove the same finite-checking result, with the same bound, for the more general \emph{shifted} Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the \emph{Lonely Vector Problem} (LVP), about sumsets of $n$ rational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with $n$ generators, a class of lattice zonotopes that we introduce. In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to $195$.

math.CO

Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture

We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes (Henze and Malikiosis, 2017), and on an upper bound for the (integer) velocities to be checked (Malikiosis, Santos and Schymura, 2024+). As a tool for the proof, we devise an algorithm for bounding the covering radius of rational lattice polytopes, based on constructing dyadic fundamental domains.

math.CO

Associahedra minimize $f$-vectors of secondary polytopes of planar point sets

Kupavskii, Volostnov, and Yarovikov have recently shown that any set of $n$ points in general position in the plane has at least as many (partial) triangulations as the convex $n$-gon. We generalize this in two directions: we show that regular triangulations are enough, and we extend the result to all regular subdivisions, graded by the dimension of their corresponding face in the secondary polytope.

math.CO

On the convex hull of integer points above the hyperbola

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $Ω(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational slopes except that instead of $n$ we have $n/Δ$ in the lower bound and by $\max\left\{Δ, n/Δ\right\}$ in the upper bound, where $Δ\in \mathbb{Z}_{>0}$ is the discriminant. We also give an algorithm that enumerates the vertices of these convex hulls in logarithmic time per vertex. One motivation for such an algorithm is the deterministic factorization of integers.

math.CO

The algorithmic Fried Potato Problem in two dimensions

Conway's Fried Potato Problem seeks to determine the best way to cut a convex body in $n$ parts by $n-1$ hyperplane cuts (with the restriction that the $i$-th cut only divides in two one of the parts obtained so far), in a way as to minimize the maxuimum of the inradii of the parts. It was shown by Bezdek and Bezdek that the solution is always attained by $n-1$ parallel cuts. But the algorithmic problem of finding the best direction for these parallel cuts remains. In this note we show that for a convex $m$-gon $P$, this direction (and hence the cuts themselves) can be found in time $O(m \log^4 m)$, which improves on a quadratic algorithm proposed by Cañete-Fernández-Márquez (DMD 2022). Our algorithm first preprocesses what we call the dome (closely related to the medial axis) of $P$ using a variant of the Dobkin-Kirkpatrick hierarchy, so that linear programs in the dome and in slices of it can be solved in polylogarithmic time.

math.MG

Realizations of multiassociahedra via rigidity

Let $Δ_k(n)$ denote the simplicial complex of $(k+1)$-crossing-free subsets of edges in $\binom{[n]}{2}$. Here $k,n\in \mathbb N$ and $n\ge 2k+1$. Jonsson (2003) proved that (neglecting the short edges that cannot be part of any $(k+1)$-crossing), $Δ_k(n)$ is a shellable sphere of dimension $k(n-2k-1)-1$, and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (2004) on subword complexes. Despite considerable effort, the only values of $(k,n)$ for which the conjecture is known to hold are $n\le 2k+3$ (Pilaud and Santos, 2012) and $(2,8)$ (Bokowski and Pilaud, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $Δ_k(n)$ as a polytope for $(k,n)\in \{(2,9), (2,10) , (3,10)\}$. We also realize it as a simplicial fan for all $n\le 13$ and arbitrary $k$, except the pairs $(3,12)$ and $(3,13)$. Finally, we also show that for $k\ge 3$ and $n\ge 2k+6$ no choice of points can realize $Δ_k(n)$ via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

math.CO

Empty simplices of large width

An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: - We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them we find five empty ones of width $11$, and none of larger width. - Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension $d$ and width growing asymptotically as $d/\operatorname{arcsinh}(1) \sim 1.1346\,d$.

math.CO

Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the $2$-dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.

math.CO

Population Graph Cross-Network Node Classification for Autism Detection Across Sample Groups

Graph neural networks (GNN) are a powerful tool for combining imaging and non-imaging medical information for node classification tasks. Cross-network node classification extends GNN techniques to account for domain drift, allowing for node classification on an unlabeled target network. In this paper we present OTGCN, a powerful, novel approach to cross-network node classification. This approach leans on concepts from graph convolutional networks to harness insights from graph data structures while simultaneously applying strategies rooted in optimal transport to correct for the domain drift that can occur between samples from different data collection sites. This blended approach provides a practical solution for scenarios with many distinct forms of data collected across different locations and equipment. We demonstrate the effectiveness of this approach at classifying Autism Spectrum Disorder subjects using a blend of imaging and non-imaging data.

cs.SI

Extension-lifting Bijections for Oriented Matroids

Extending the notion of geometric bijections for regular matroids, introduced by the first and third author with Matthew Baker, we describe a family of bijections between bases of an oriented matroid and special orientations. These bijections are specified by a pair of circuit and cocircuit signatures coming respectively from a generic single-element lifting and extension. We then characterize generic single-element liftings and extensions using these bijections. We also explain the relation of our work with the works of Gioan--Las Vergnas and Ding. Some implications in oriented matroid programming and oriented matroid triangulations are also discussed.

math.CO

The structure of monotone blow-ups in symplectic toric geometry and a question of McDuff

Monotone polytopes, also known as smooth reflexive polytopes, are the polytopes associated to monotone symplectic toric manifolds and Gorenstein Fano toric varieties. We first show that the only monotone polytopes admitting blow-ups at vertices are the simplex and the result of a codimension-two blow-up in it (this is the polyhedral version of a result of Bonavero from 2002). Then we show that the $n$-simplex admits disjoint blow-ups at faces if and only if the faces are disjoint and have dimensions adding up to $n-1$ or $n-2$. These results answer a question posed by Dusa McDuff in 2011.

math.SG

Many regular triangulations and many polytopes

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an intermediate step, we show that certain neighborly polytopes (such as particular realizations of cyclic polytopes) have at least $(n!)^{ \lfloor(d-1)/2\rfloor \pm o(1)}$ regular triangulations.

math.CO

Celebrating Loday's Associahedron

We survey Jean-Louis Loday's vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry and algebra. We present in particular four topics were it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal.

math.CO

Bar-and-joint rigidity on the moment curve coincides with cofactor rigidity on a conic

We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the $C^{d-2}_{d-1}$-cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality. We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.

math.CO

Multitriangulations and tropical Pfaffians

The $k$-associahedron $Ass_k(n)$ is the simplicial complex of $(k+1)$-crossing-free subgraphs of the complete graph with vertices on a circle. Its facets are called $k$-triangulations. We explore the connection of $Ass_k(n)$ with the Pfaffian variety $Pf_k(n)\subset {\mathbb K}^{\binom{[n]}2}$ of antisymmetric matrices of rank $\le 2k$. First, we characterize the Gröbner cone $Grob_k(n)\subset{\mathbb R}^{\binom{[n]}2}$ producing as initial ideal of $I(Pf_k(n))$ the Stanley-Reisner ideal of $Ass_k(n)$ (that is, the monomial ideal generated by $(k+1)$-crossings). This implies that $k$-triangulations are bases in the algebraic matroid of $Pf_k(n)$, a matroid closely related to low-rank completion of antisymmetric matrices. We then look at the tropicalization of $Pf_k(n)$ and show that $Ass_k(n)$ embeds naturally as the intersection of $\operatorname{trop}(Pf_k(n))$ and $Grob_k(n)$, and is contained in the totally positive part $\operatorname{trop}^+( Pf_k(n))$ of it. We show that for $k=1$ and for each triangulation $T$ of the $n$-gon, the projection of this embedding of $Ass_k(n)$ to the $n-3$ coordinates corresponding to diagonals in $T$ gives a complete polytopal fan, realizing the associahedron. This fan is linearly isomorphic to the $\mathbf g$-vector fan of the cluster algebra of type $A$, shown to be polytopal by Hohlweg, Pilaud and Stella in (2018).

math.CO