819 research outputs found
Three-dimensional compressible stability-transition calculations using the spatial theory
The e(exp n)-method is employed with the spatial amplification theory to compute the onset of transition on a swept wing tested in transonic cryogenic flow conditions. Two separate eigenvalue formulations are used. One uses the saddle-point method and the other assumes that the amplification vector is normal to the leading edge. Comparisons of calculated results with experimental data show that both formulations give similar results and indicate that the wall temperature has a rather strong effect on the value of the n factor
Dissecting Supervised Contrastive Learning
Minimizing cross-entropy over the softmax scores of a linear map composed
with a high-capacity encoder is arguably the most popular choice for training
neural networks on supervised learning tasks. However, recent works show that
one can directly optimize the encoder instead, to obtain equally (or even more)
discriminative representations via a supervised variant of a contrastive
objective. In this work, we address the question whether there are fundamental
differences in the sought-for representation geometry in the output space of
the encoder at minimal loss. Specifically, we prove, under mild assumptions,
that both losses attain their minimum once the representations of each class
collapse to the vertices of a regular simplex, inscribed in a hypersphere. We
provide empirical evidence that this configuration is attained in practice and
that reaching a close-to-optimal state typically indicates good generalization
performance. Yet, the two losses show remarkably different optimization
behavior. The number of iterations required to perfectly fit to data scales
superlinearly with the amount of randomly flipped labels for the supervised
contrastive loss. This is in contrast to the approximately linear scaling
previously reported for networks trained with cross-entropy.Comment: ICML 2021 camera ready versio
Joint and individual analysis of breast cancer histologic images and genomic covariates
A key challenge in modern data analysis is understanding connections between
complex and differing modalities of data. For example, two of the main
approaches to the study of breast cancer are histopathology (analyzing visual
characteristics of tumors) and genetics. While histopathology is the gold
standard for diagnostics and there have been many recent breakthroughs in
genetics, there is little overlap between these two fields. We aim to bridge
this gap by developing methods based on Angle-based Joint and Individual
Variation Explained (AJIVE) to directly explore similarities and differences
between these two modalities. Our approach exploits Convolutional Neural
Networks (CNNs) as a powerful, automatic method for image feature extraction to
address some of the challenges presented by statistical analysis of
histopathology image data. CNNs raise issues of interpretability that we
address by developing novel methods to explore visual modes of variation
captured by statistical algorithms (e.g. PCA or AJIVE) applied to CNN features.
Our results provide many interpretable connections and contrasts between
histopathology and genetics
Thalamocortical Connectivity Correlates with Phenotypic Variability in Dystonia
Dystonia is a brain disorder characterized by abnormal involuntary movements without defining neuropathological changes. The disease is often inherited as an autosomal-dominant trait with incomplete penetrance. Individuals with dystonia, whether inherited or sporadic, exhibit striking phenotypic variability, with marked differences in the somatic distribution and severity of clinical manifestations. In the current study, we used magnetic resonance diffusion tensor imaging to identify microstructural changes associated with specific limb manifestations. Functional MRI was used to localize specific limb regions within the somatosensory cortex. Microstructural integrity was preserved when assessed in subrolandic white matter regions somatotopically related to the clinically involved limbs, but was reduced in regions linked to clinically uninvolved (asymptomatic) body areas. Clinical manifestations were greatest in subjects with relatively intact microstructure in somatotopically relevant white matter regions. Tractography revealed significant phenotype-related differences in the visualized thalamocortical tracts while corticostriatal and corticospinal pathways did not differ between groups. Cerebellothalamic microstructural abnormalities were also seen in the dystonia subjects, but these changes were associated with genotype, rather than with phenotypic variation. The findings suggest that the thalamocortical motor system is a major determinant of dystonia phenotype. This pathway may represent a novel therapeutic target for individuals with refractory limb dystonia
Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction
We study a singular-limit problem arising in the modelling of chemical
reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck
convection-diffusion equation with a double-well convection potential. This
potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the
solution concentrates onto the two wells, resulting into a limiting system that
is a pair of ordinary differential equations for the density at the two wells.
This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM
Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear
structure of the equation. In this paper we re-prove the result by using solely
the Wasserstein gradient-flow structure of the system. In particular we make no
use of the linearity, nor of the fact that it is a second-order system. The
first key step in this approach is a reformulation of the equation as the
minimization of an action functional that captures the property of being a
curve of maximal slope in an integrated form. The second important step is a
rescaling of space. Using only the Wasserstein gradient-flow structure, we
prove that the sequence of rescaled solutions is pre-compact in an appropriate
topology. We then prove a Gamma-convergence result for the functional in this
topology, and we identify the limiting functional and the differential equation
that it represents. A consequence of these results is that solutions of the
{\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference
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