22,499 research outputs found
Rank-based linkage I: triplet comparisons and oriented simplicial complexes
Rank-based linkage is a new tool for summarizing a collection of objects
according to their relationships. These objects are not mapped to vectors, and
``similarity'' between objects need be neither numerical nor symmetrical. All
an object needs to do is rank nearby objects by similarity to itself, using a
Comparator which is transitive, but need not be consistent with any metric on
the whole set. Call this a ranking system on . Rank-based linkage is applied
to the -nearest neighbor digraph derived from a ranking system. Computations
occur on a 2-dimensional abstract oriented simplicial complex whose faces are
among the points, edges, and triangles of the line graph of the undirected
-nearest neighbor graph on . In steps it builds an
edge-weighted linkage graph where
is called the in-sway between objects and . Take to be
the links whose in-sway is at least , and partition into components of
the graph , for varying . Rank-based linkage is a
functor from a category of out-ordered digraphs to a category of partitioned
sets, with the practical consequence that augmenting the set of objects in a
rank-respectful way gives a fresh clustering which does not ``rip apart`` the
previous one. The same holds for single linkage clustering in the metric space
context, but not for typical optimization-based methods. Open combinatorial
problems are presented in the last section.Comment: 37 pages, 12 figure
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Ensuring Access to Safe and Nutritious Food for All Through the Transformation of Food Systems
Quantum Mechanics Lecture Notes. Selected Chapters
These are extended lecture notes of the quantum mechanics course which I am
teaching in the Weizmann Institute of Science graduate physics program. They
cover the topics listed below. The first four chapter are posted here. Their
content is detailed on the next page. The other chapters are planned to be
added in the coming months.
1. Motion in External Electromagnetic Field. Gauge Fields in Quantum
Mechanics.
2. Quantum Mechanics of Electromagnetic Field
3. Photon-Matter Interactions
4. Quantization of the Schr\"odinger Field (The Second Quantization)
5. Open Systems. Density Matrix
6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation
7. Mean Field Approaches for Many Body Systems -- Fermions and Boson
Tonelli Approach to Lebesgue Integration
Leonida Tonelli devised an interesting and efficient method to introduce the
Lebesgue integral. The details of this method can only be found in the original
Tonelli paper and in an old italian course and solely for the case of the
functions of one variable. We believe that it is woth knowing this method and
here we present a complete account for functions of every number of variables
Properties of a model of sequential random allocation
Probabilistic models of allocating shots to boxes according to a certain probability distribution have commonly been used for processes involving agglomeration. Such processes are of interest in many areas of research such as ecology, physiology, chemistry and genetics. Time could be incorporated into the shots-and-boxes model by considering multiple layers of boxes through which the shots move, where the layers represent the passing of time. Such a scheme with multiple layers, each with a certain number of occupied boxes is naturally associated with a random tree. It lends itself to genetic applications where the number of ancestral lineages of a sample changes through the generations. This multiple-layer scheme also allows us to explore the difference in the number of occupied boxes between layers, which gives a measure of how quickly merges are happening. In particular, results for the multiple-layer scheme corresponding to those known for a single-layer scheme, where, under certain conditions, the limiting distribution of the number of occupied boxes is either Poisson or normal, are derived. To provide motivation and demonstrate which methods work well, a detailed study of a small, finite example is provided. A common approach for establishing a limiting distribution for a random variable of interest is to first show that it can be written as a sum of independent Bernoulli random variables as this then allows us to apply standard central limit theorems. Additionally, it allows us to, for example, provide an upper bound on the distance to a Poisson distribution. One way of showing that a random variable can be written as a sum of independent Bernoulli random variables is to show that its probability generating function (p.g.f.) has all real roots. Various methods are presented and considered for proving the p.g.f. of the number of occupied boxes in any given layer of the scheme has all real roots. By considering small finite examples some of these methods could be ruled out for general N. Finally, the scheme for general N boxes and n shots is considered, where again a uniform allocation of shots is used. It is shown that, under certain conditions, the distribution of the number of occupied boxes tends towards either a normal or Poisson limit. Equivalent results are also demonstrated for the distribution of the difference in the number of occupied boxes between consecutive layers
Isotopic piecewise affine approximation of algebraic or varieties
We propose a novel sufficient condition establishing that a piecewise affine
variety has the same topology as a variety of the sphere defined
by positively homogeneous functions. This covers the case of
varieties in the projective space . We prove that this condition
is sufficient in the case of codimension one and arbitrary dimension. We
describe an implementation working for homogeneous polynomials in arbitrary
dimension and codimension and give experimental evidences that our condition
might still be sufficient in codimension greater than one
Nonparametric Two-Sample Test for Networks Using Joint Graphon Estimation
This paper focuses on the comparison of networks on the basis of statistical
inference. For that purpose, we rely on smooth graphon models as a
nonparametric modeling strategy that is able to capture complex structural
patterns. The graphon itself can be viewed more broadly as density or intensity
function on networks, making the model a natural choice for comparison
purposes. Extending graphon estimation towards modeling multiple networks
simultaneously consequently provides substantial information about the
(dis-)similarity between networks. Fitting such a joint model - which can be
accomplished by applying an EM-type algorithm - provides a joint graphon
estimate plus a corresponding prediction of the node positions for each
network. In particular, it entails a generalized network alignment, where
nearby nodes play similar structural roles in their respective domains. Given
that, we construct a chi-squared test on equivalence of network structures.
Simulation studies and real-world examples support the applicability of our
network comparison strategy.Comment: 25 pages, 6 figure
Four Lectures on the Random Field Ising Model, Parisi-Sourlas Supersymmetry, and Dimensional Reduction
Numerical evidence suggests that the Random Field Ising Model loses
Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4
and 5 dimensions, while a related model of branched polymers retains these
features in any . These notes give a leisurely introduction to a recent
theory, developed jointly with A. Kaviraj and E. Trevisani, which aims to
explain these facts. Based on the lectures given in Cortona and at the IHES in
2022.Comment: 55 pages, 11 figures; v2 - minor changes, mentioned forthcoming work
by Fytas et a
Structured Dynamic Pricing: Optimal Regret in a Global Shrinkage Model
We consider dynamic pricing strategies in a streamed longitudinal data set-up
where the objective is to maximize, over time, the cumulative profit across a
large number of customer segments. We consider a dynamic probit model with the
consumers' preferences as well as price sensitivity varying over time. Building
on the well-known finding that consumers sharing similar characteristics act in
similar ways, we consider a global shrinkage structure, which assumes that the
consumers' preferences across the different segments can be well approximated
by a spatial autoregressive (SAR) model. In such a streamed longitudinal
set-up, we measure the performance of a dynamic pricing policy via regret,
which is the expected revenue loss compared to a clairvoyant that knows the
sequence of model parameters in advance. We propose a pricing policy based on
penalized stochastic gradient descent (PSGD) and explicitly characterize its
regret as functions of time, the temporal variability in the model parameters
as well as the strength of the auto-correlation network structure spanning the
varied customer segments. Our regret analysis results not only demonstrate
asymptotic optimality of the proposed policy but also show that for policy
planning it is essential to incorporate available structural information as
policies based on unshrunken models are highly sub-optimal in the
aforementioned set-up.Comment: 34 pages, 5 figure
mSPD-NN: A Geometrically Aware Neural Framework for Biomarker Discovery from Functional Connectomics Manifolds
Connectomics has emerged as a powerful tool in neuroimaging and has spurred
recent advancements in statistical and machine learning methods for
connectivity data. Despite connectomes inhabiting a matrix manifold, most
analytical frameworks ignore the underlying data geometry. This is largely
because simple operations, such as mean estimation, do not have easily
computable closed-form solutions. We propose a geometrically aware neural
framework for connectomes, i.e., the mSPD-NN, designed to estimate the geodesic
mean of a collections of symmetric positive definite (SPD) matrices. The
mSPD-NN is comprised of bilinear fully connected layers with tied weights and
utilizes a novel loss function to optimize the matrix-normal equation arising
from Fr\'echet mean estimation. Via experiments on synthetic data, we
demonstrate the efficacy of our mSPD-NN against common alternatives for SPD
mean estimation, providing competitive performance in terms of scalability and
robustness to noise. We illustrate the real-world flexibility of the mSPD-NN in
multiple experiments on rs-fMRI data and demonstrate that it uncovers stable
biomarkers associated with subtle network differences among patients with
ADHD-ASD comorbidities and healthy controls.Comment: Accepted into IPMI 202
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