arXiv research

Curve shortening flow increases annulus modulus.

We apply Murasugi-Tristram inequality to real algebraic curves of odd degree on $RP^2$ with a deep nest, i.e. a nest of the depth $k-1$ where $2k+1$ is the degree. For such curves, the ingredients of the Murasugi-Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus li…

The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitti…

A new method for efficient nested Monte Carlo simulations in financial modeling.

We construct Peano curves $γ: [0,\infty) \to \mathbb{R}^2$ whose "footprints" $γ([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent to a common continuous line field on the punctured plane $\mathbb{R}^2 \setminus \{γ(0)\}$. Moreover, these boundaries can be taken $C^\infty$-close to any prescribed smooth family…

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the 'nested complex…

This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theori…

Bayesian network models with latent variables are widely used in statistics and machine learning. In this paper we provide a complete algebraic characterization of Bayesian network models with latent variables when the observed variables are discrete and no assumption is made about the state-space of the latent variabl…

Study compares MCMC and nested sampling for high-dimensional physics problems.

Study cobordisms of nested manifolds and their invariants.

Quantum algorithm speeds up nested expectation estimation by nearly quadratically.

Improved nested simulation for financial risk measurement.

Scalable tools for nested optimization in deep learning.

Paper proposes a new estimator for nested expectations with faster convergence.

Unified SGD method improves convergence for nested optimization problems.

Let R be an o-minimal expansion of the real field, and let L(R) be the language consisting of all nested Rolle leaves over R. We call a set nested subpfaffian over R if it is the projection of a boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we …

Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistica…

Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.

Develops a method for learning proposals in nested importance samplers.

Nested model averaging improves high-dimensional linear regression performance.

Gradient-guided nested sampling improves posterior inference efficiency.

Nested Slice Sampling accelerates Nested Sampling for GPU acceleration.

New discrete cobordism category for nested manifolds and relations to algebraic structures.

Nested sampling improved for arbitrary priors.

Study of loops in sums of Laplace eigenfunctions on surfaces.

Enhances Bayesian model selection for high-dimensional problems.

Nested learning improves model performance on multi-granular tasks.

NEST optimizes deep learning training by placing devices efficiently across networks and memory.

We propose doubly nested network(DNNet) where all neurons represent their own sub-models that solve the same task. Every sub-model is nested both layer-wise and channel-wise. While nesting sub-models layer-wise is straight-forward with deep-supervision as proposed in \cite{xie2015holistically}, channel-wise nesting has…

New methods for estimating complex causal effects in econometrics.

We classify the triples $H \subset K \subset G$ of nested compact Lie groups which satisfy the "positive triple" condition that was shown by the second author to ensure that $G/H$ admits a metric with quasi-positive curvature. A few new examples of spaces that admit quasi-positively curved metrics emerge from this clas…

Throughout science and technology, receiver operating characteristic (ROC) curves and associated area under the curve (AUC) measures constitute powerful tools for assessing the predictive abilities of features, markers and tests in binary classification problems. Despite its immense popularity, ROC analysis has been su…

We formalize the notion of nesting probabilistic programming queries and investigate the resulting statistical implications. We demonstrate that while query nesting allows the definition of models which could not otherwise be expressed, such as those involving agents reasoning about other agents, existing systems take …

Sharp lower bound on GHHs' representation power of CPWL functions.

We develop nested automatic differentiation (AD) algorithms for exact inference and learning in integer latent variable models. Recently, Winner, Sujono, and Sheldon showed how to reduce marginalization in a class of integer latent variable models to evaluating a probability generating function which contains many leve…

There is an increasing interest in estimating expectations outside of the classical inference framework, such as for models expressed as probabilistic programs. Many of these contexts call for some form of nested inference to be applied. In this paper, we analyse the behaviour of nested Monte Carlo (NMC) schemes, for w…

Paper tackles robust model training with a new stochastic algorithm.

EENNs improve inference efficiency but need nested prediction sets for reliable uncertainty estimates.

Paper tackles robust optimization under uncertainty using nested distance.

Study on MHD equilibria on curved spaces without symmetries.

New formulas compare total mean curvatures of nested hypersurfaces.

New unbiased gradient estimators for complex optimization problems.

New estimator reduces nested expectation estimation costs.

Develops ML tool for macroeconomic forecasting with clear interpretations.

A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively splits the set of classes into two subsets, and trains a binary classifier to distinguish between each subset. Even though ensembles of nested dichotomies with random structure…

Study geodesics on nested non-holonomic systems.

We show that deliberately introducing a nested simulation stage can lead to significant variance reductions when comparing two stopping times by Monte Carlo. We derive the optimal number of nested simulations and prove that the algorithm is remarkably robust to misspecifications of this number. The method is applied to…