arXiv research

The paper interprets financial markets as crowds during booms and busts.

Despite all the success that deep neural networks have seen in classifying certain datasets, the challenge of finding optimal solutions that generalize still remains. In this paper, we propose the Boundary Optimizing Network (BON), a new approach to generalization for deep neural networks when used for supervised learn…

Improved selection of best outputs from LLMs for better accuracy.

Improved inference-time alignment using Best-of-N and smoothing.

Best-of-Majority improves inference performance in Pass@$k$ settings.

Two methods for model adaptation compared; fine-tuning outperforms Best-of-N in realizable settings.

The study examines how verifier imperfections impact test-time scaling techniques.

New method predicts and optimizes test-time scaling for LLMs.

The application to search ranking is one of the biggest machine learning success stories at Airbnb. Much of the initial gains were driven by a gradient boosted decision tree model. The gains, however, plateaued over time. This paper discusses the work done in applying neural networks in an attempt to break out of that …

Proves non-existence of metrics with positive curvature for certain connected sums.

En nous basant sur les résultats d'Arthur annoncés dans \cite[§30]{Arthur} nous démontrons les conjectures énoncées dans \cite{IMRN,BC,SMF} dans le cas des groupes orthogonaux à l'exclusion des groupes de type ${}^6 - D_4$. En ce qui concerne ces derniers, nous annonçons la démonstration -- encore en préparation - que …

Nous considérons un espace topologique qui est localement isomorphe au quotient de R^k par l'action d'un groupe discret et nous l'appelons quasi-variété de dimension k. Les quasi-variétés généralisent les variétés et les V-variétés et représentent le cadre naturel pour la réduction symplectique par rapport à l'action i…

Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric spac…

LES reduces over-exploration in LSO, improving solution quality.

New invariant prevents minimal submanifolds in curved spaces.

We give an upper bound for the rank $r$ of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if $r=2^a\cdot b$ with $b$ odd, then $r\le 9$ for $a=0$, $r\le 10$ for $a=1$, $r\le 12$ for $a=2$ and $r\le 16$ for $a\ge 3$. Moreover, we describe t…

We inspect the BNSR-invariants $Σ^m(P_n)$ of the pure braid groups $P_n$, using Morse theory. The BNS-invariants $Σ^1(P_n)$ were previously computed by Koban, McCammond and Meier. We prove that for any $3\le m\le n$, the inclusion $Σ^{m-2}(P_n)\subseteq Σ^{m-3}(P_n)$ is proper, but $Σ^\infty(P_n)=Σ^{n-2}(P_n)$. We writ…

Homotopy equivalence found between Milnor-Lê fibers of specific singularities.

PCR-LE achieves optimal rates for nonparametric regression over Sobolev spaces.

After results by the author (1980, 1981), and by Vinberg (1981), finiteness of the number of maximal arithmetic reflection groups in Lobachevsky spaces was not known in dimensions $2\le n\le 9$ only. Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol. Here…

Let $(M^n,g)$, $n \ge 4$, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\eps = \eps (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies $l \le s \le L$ and the Einstein tensor satisfies $$ | Ric - \fr…

Using elementary comparison geometry, we prove: Let $(M,g)$ be a simply-connected complete Riemannian manifold of dimension $\ge 3$. Suppose that the sectional curvature $K$ satisfies $-1-s(r) \le K \le -1$, where $r$ denotes distance to a fixed point in $M$. If $\lim_{r \rt \infty} e^{2r}s(r) =0$, then $(M,g)$ has to…

Study correlations of spectral lengths and displacements in higher rank groups.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Eu…

This is a translation of Yves Benoist's "Actions propres sur les espaces homogènes réductifs", Ann. of Math., 144:315-347, 1996.

Study compares survival analysis algorithms with missing data methods.

New invariants show stronger virtual knot sets.

The paper proves rigidity for certain product spaces and bounds for band widths.

New proof confirms petal number for torus knots without modular condition.

We consider a real analytic map $F=(f_1,...,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)$, $2 \le k \le n-1$, that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic $f_I=(f_{i_1},...,f_{i_l}) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^l,0)$, induced from $F…

The paper bounds growth indicator functions for discrete subgroups in algebraic groups.

Dans les années 1940-1970, Alexandrov et l'"École de Leningrad" ont développé une théorie très riche des surfaces singulières. Il s'agit de surfaces topologiques, munie d'une métrique intrinsèque pour laquelle on peut définir une notion de courbure, qui est une mesure de Radon. Cette classe de surfaces a de bonnes prop…

A new framework for robust transfer learning that avoids negative transfer in domains with unequal information.

We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used in modeling homologous DNA recombination. We show that the genus ranges are sets …

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to S^m$ (respectively, $S^p\to S^m$) up to smooth isotopy. Denote by $LM^m_{p,q}$ the …

The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim …

We prove that for 4-manifolds $M$ with residually finite fundamental group and non-spin universal covering $\Wi M$, the inequality $\dim_{mc}\Wi M\le 3$ implies the inequality $\dim_{mc}\Wi M\le 2$.

Proves inequalities for hypersurfaces in the sphere, solving a long-standing problem.

We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)

For any hyperbolic twist knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ satisfies the inequality $0\le r \le 4$.

We prove that each non-separable completely metrizable convex subset of a Frechet space is homeomorphic to a Hilbert space. This resolves an old (more than 30 years) problem of infinite-dimensional topology. Combined with the topological classification of separable convex sets due to Klee, Dobrowoslki and Torunczyk, th…

The paper generalizes the Hausdorff dimension of limit sets for self-joinings of hyperbolic groups.

We prove that any real analytic strictly pseudoconvex CR 3-manifold is the boundary (at infinity) of a unique selfdual Einstein metric defined in a neighborhood. The proof uses a new construction of twistor space based on singular rational curves.

We present a new explicit formula for the $m$-th Bernoulli number $B_m$, which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker formula for $B_m$. We give two proofs of our formula. One is analytic and uses a certain fun…

Method tackles uncertainty in reward models for LLMs from heterogeneous human feedback.

It is proved that for a product action of $(\mathbb Z_p)^k$ on a product of (mod p) homology spheres $N^{n_1}\times...\times N^{n_k}$, where all $n_i$'s are assumed to be odd if $p$ is odd, and any continuous map $f\colon N^{n_1}\times...\times N^{n_k}\to \mathbb R^m$ the set $A(f)=\{x\in N^{n_1}\times...\times N^{n_k}…

We show that the resulting manifold by $r$-surgery on the knot $5_2$, which is the two-bridge knot corresponding to the rational number 3/7, has left-orderable fundamental group if the slope $r$ satisfies $0\le r\le 4$.

We study the rigidity and flexibility of symplectic embeddings of simple shapes. It is first proved that under the condition $r_n^2 \le 2 r_1^2$ the symplectic ellipsoid $E(r_1, ..., r_n)$ with radii $r_1 \le ... \le r_n$ does not embed in a ball of radius strictly smaller than $r_n$. We then use symplectic folding to …