18 research outputs found
Computing the elementary symmetric polynomials of the multiplier spectra of the maps ²+
Let f be a complex quadratic rational map. The ith elementary symmetric polynomial of the formal n multiplier spectra of f is denoted &sigmai(n)(f). The values of these polynomials are invariant under conjugation by the projective linear group and are interesting to the study of the moduli space of quadratic rational maps. For every positive integer n and i in the appropriate range, &sigmai(n)(f) is in Z[&sigma1, &sigma2] where &sigma1, &sigma2 are &sigma1(1)(f, &sigma2(1)(f), respectively. Despite this, the &sigmai(n)(f) are difficult to compute. By restricting our focus to the family of quadratic polynomials z2+c, computations become simpler. We determine an upper bound for the degrees of the &sigmai(n) for the maps of the form z2+c by arguing in terms of the growth rates of their periodic points and corresponding multipliers. We also include computations of the forms of the &sigmai(n) for n =2,…,6 for these maps
Neural frames: A Tool for Studying the Tangent Bundles Underlying Image Datasets and How Deep Learning Models Process Them
The assumption that many forms of high-dimensional data, such as images,
actually live on low-dimensional manifolds, sometimes known as the manifold
hypothesis, underlies much of our intuition for how and why deep learning
works. Despite the central role that they play in our intuition, data manifolds
are surprisingly hard to measure in the case of high-dimensional, sparsely
sampled image datasets. This is particularly frustrating since the capability
to measure data manifolds would provide a revealing window into the inner
workings and dynamics of deep learning models. Motivated by this, we introduce
neural frames, a novel and easy to use tool inspired by the notion of a frame
from differential geometry. Neural frames can be used to explore the local
neighborhoods of data manifolds as they pass through the hidden layers of
neural networks even when one only has a single datapoint available. We present
a mathematical framework for neural frames and explore some of their
properties. We then use them to make a range of observations about how modern
model architectures and training routines, such as heavy augmentation and
adversarial training, affect the local behavior of a model.Comment: 21 page
In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?
It is often said that a deep learning model is "invariant" to some specific
type of transformation. However, what is meant by this statement strongly
depends on the context in which it is made. In this paper we explore the nature
of invariance and equivariance of deep learning models with the goal of better
understanding the ways in which they actually capture these concepts on a
formal level. We introduce a family of invariance and equivariance metrics that
allows us to quantify these properties in a way that disentangles them from
other metrics such as loss or accuracy. We use our metrics to better understand
the two most popular methods used to build invariance into networks: data
augmentation and equivariant layers. We draw a range of conclusions about
invariance and equivariance in deep learning models, ranging from whether
initializing a model with pretrained weights has an effect on a trained model's
invariance, to the extent to which invariance learned via training can
generalize to out-of-distribution data.Comment: To appear at NeurIPS 202
Convolutional networks inherit frequency sensitivity from image statistics
It is widely acknowledged that trained convolutional neural networks (CNNs)
have different levels of sensitivity to signals of different frequency. In
particular, a number of empirical studies have documented CNNs sensitivity to
low-frequency signals. In this work we show with theory and experiments that
this observed sensitivity is a consequence of the frequency distribution of
natural images, which is known to have most of its power concentrated in
low-to-mid frequencies. Our theoretical analysis relies on representations of
the layers of a CNN in frequency space, an idea that has previously been used
to accelerate computations and study implicit bias of network training
algorithms, but to the best of our knowledge has not been applied in the domain
of model robustness.Comment: Comments welcome
ColMix -- A Simple Data Augmentation Framework to Improve Object Detector Performance and Robustness in Aerial Images
In the last decade, Convolutional Neural Network (CNN) and transformer based
object detectors have achieved high performance on a large variety of datasets.
Though the majority of detection literature has developed this capability on
datasets such as MS COCO, these detectors have still proven effective for
remote sensing applications. Challenges in this particular domain, such as
small numbers of annotated objects and low object density, hinder overall
performance. In this work, we present a novel augmentation method, called
collage pasting, for increasing the object density without a need for
segmentation masks, thereby improving the detector performance. We demonstrate
that collage pasting improves precision and recall beyond related methods, such
as mosaic augmentation, and enables greater control of object density. However,
we find that collage pasting is vulnerable to certain out-of-distribution
shifts, such as image corruptions. To address this, we introduce two simple
approaches for combining collage pasting with PixMix augmentation method, and
refer to our combined techniques as ColMix. Through extensive experiments, we
show that employing ColMix results in detectors with superior performance on
aerial imagery datasets and robust to various corruptions