A fast shape transformation using a phase-field model

Abstract Neural networks functions are supposed to be able to encode the desired solution of an inverse problem very efficiently. In this paper, we consider the problem of solving linear inverse problems with neural network coders. First we establish some correspondences of this formulation with existing concepts in regularization theory, in particular with state space regularization, operator decomposition and iterative regularization methods. A Gauss–Newton method is suitable for solving encoded linear inverse problems, which is supported by a local convergence result. The convergence studies, however, are not complete, and are based on a conjecture on linear independence of activation functions and its derivatives. Some numerical experiments are presented to support the theoretical findings. 

Gauss–Newton method for solving linear inverse problems with neural network coders

Deep learning models extract, before a final classification layer, features or patterns which are key for their unprecedented advantageous performance However, the process of complex nonlinear feature extraction is not well understood, a major reason why interpretation, adversarial robustness, and generalization of deep neural nets are all open research problems In this paper, we use wavelet transformation and spectral methods to analyze the contents of image classification datasets, extract specific patterns from the datasets and find the associations between patterns and classes We show that each image can be written as the summation of a finite number of rank-1 patterns in the wavelet space, providing a low rank approximation that captures the structures and patterns essential for learning Regarding the studies on memorization vs learning, our results clearly reveal disassociation of patterns from classes, when images are randomly labeled Our method can be used as a pattern recognition approach to understand and interpret learnability of these datasets It may also be used for gaining insights about the features and patterns that deep classifiers learn from the datasets

Using Wavelets and Spectral Methods to Study Patterns in Image-Classification Datasets.

The success of deep neural networks in image classiﬁcation and learning can be partly attributed to the features they extract from images. It is often speculated about the properties of a low-dimensional manifold that models extract and learn from images. However, there is not suf-ﬁcient understanding about this low-dimensional space based on theory or empirical evidence. For image classiﬁcation models, their last hidden layer is the one where images of each class is separated from other classes and it also has the least number of features. Here, we develop methods and formulations to study that feature space for any model. We study the partitioning of the domain in feature space, identify regions guaranteed to have certain classiﬁcations, and investigate its implications for the pixel space. We observe that geometric arrangements of decision boundaries in feature space is signiﬁcantly different compared to pixel space, providing insights about adversarial vulnerabilities, image morphing, extrapolation, ambiguity in classiﬁcation, and the mathematical understanding of image classiﬁcation models.

Decision boundaries and convex hulls in the feature space that deep learning functions learn from images

Cubic splines are a classical tool for higher order interpolation of points in Euclidean space known to minimize the integral of the squared acceleration along the interpolation path. This paper transfers this method to the smooth interpolation of key frames in the space of images. To this end the metamorphosis model based on a simultaneous transport of image intensities and a modulation of intensities along motion trajectories is generalized. The proposed spline energy combines quadratic functionals of the Eulerian motion acceleration and of the second material derivative representing an acceleration in the change of intensities along motion paths. A variational time discretization of this spline model is proposed and the convergence to a suitably relaxed time continuous model is discussed using the tool of \(\varGamma \)-convergence. In particular, this also allows to establish the existence of an optimal spline path interpolating given key frame images. The spatial discretization is based on a finite difference and a stable spline interpolation. A variety of numerical examples demonstrates the robustness and versatility of the proposed method for real images using a variant of the iPALM algorithm for the minimization of the fully discrete energy functional.

Splines for Image Metamorphosis

This paper combines image metamorphosis with deep features. To this end, images are considered as maps into a high-dimensional feature space and a structure-sensitive, anisotropic flow regularization is incorporated in the metamorphosis model proposed by Miller and Younes (Int J Comput Vis 41(1):61–84, 2001) and Trouve and Younes (Found Comput Math 5(2):173–198, 2005). For this model, a variational time discretization of the Riemannian path energy is presented and the existence of discrete geodesic paths minimizing this energy is demonstrated. Furthermore, convergence of discrete geodesic paths to geodesic paths in the time continuous model is investigated. The spatial discretization is based on a finite difference approximation in image space and a stable spline approximation in deformation space; the fully discrete model is optimized using the iPALM algorithm. Numerical experiments indicate that the incorporation of semantic deep features is superior to intensity-based approaches.

/pdf/image-morphing-in-deep-feature-spaces-theory-and-58mwhpu21o.pdf

Image Morphing in Deep Feature Spaces: Theory and Applications.

This paper combines image metamorphosis with deep features. To this end, images are considered as maps into a high-dimensional feature space and a structure-sensitive, anisotropic flow regularization is incorporated in the metamorphosis model proposed by Miller, Trouve, Younes and coworkers. For this model a variational time discretization of the Riemannian path energy is presented and the existence of discrete geodesic paths minimizing this energy is demonstrated. Furthermore, convergence of discrete geodesic paths to geodesic paths in the time continuous model is investigated. The spatial discretization is based on a finite difference approximation in image space and a stable spline approximation in deformation space, the fully discrete model is optimized using the iPALM algorithm. Numerical experiments indicate that the incorporation of semantic deep features is superior to intensity-based approaches.

pdf/image-morphing-in-deep-feature-spaces-theory-and-34oefen75i.pdf

Image Morphing in Deep Feature Spaces: Theory and Applications

Autoencoders are a widespread tool in machine learning to transform high-dimensional data into a lower-dimensional representation which still exhibits the essential characteristics of the input. The encoder provides an embedding from the input data manifold into a latent space which may then be used for further processing. For instance, learning interpolation on the manifold may be simplified via the new manifold representation in latent space. The efficiency of such further processing heavily depends on the regularity and structure of the embedding. In this article, the embedding into latent space is regularized via a loss function that promotes an as isometric and as flat embedding as possible. The required training data comprises pairs of nearby points on the input manifold together with their local distance and their local Frechet average. This regularity loss functional even allows to train the encoder on its own. The loss functional is computed via a Monte Carlo integration which is shown to be consistent with a geometric loss functional defined directly on the embedding map. Numerical tests are performed using image data that encodes different data manifolds. The results show that smooth manifold embeddings in latent space are obtained. These embeddings are regular enough such that interpolation between not too distant points on the manifold is well approximated by linear interpolation in latent space.

/pdf/learning-low-bending-and-low-distortion-manifold-embeddings-2bzn92tdph.pdf

Learning low bending and low distortion manifold embeddings

Autoencoders are a widespread tool in machine learning to transform high-dimensional data into a lowerdimensional representation which still exhibits the essential characteristics of the input. The encoder provides an embedding from the input data manifold into a latent space which may then be used for further processing. For instance, learning interpolation on the manifold may be simplified via the new manifold representation in latent space. The efficiency of such further processing heavily depends on the regularity and structure of the embedding. In this article, the embedding into latent space is regularized via a loss function that promotes an as isometric and as flat embedding as possible. The required training data comprises pairs of nearby points on the input manifold together with their local distance and their local Frechet average. This regularity loss functional even allows to train the encoder on its own. The loss functional is computed via a Monte Carlo integration which is shown to be consistent with a geometric loss functional defined directly on the embedding map. Numerical tests are performed using image data that encodes different data manifolds. The results show that smooth manifold embeddings in latent space are obtained. These embeddings are regular enough such that interpolation between not too distant points on the manifold is well approximated by linear interpolation in latent space.

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Image Morphing in Deep Feature Spaces: Theory and Applications.

Image Morphing in Deep Feature Spaces: Theory and Applications

Learning low bending and low distortion manifold embeddings

Splines for Image Metamorphosis

Learning low bending and low distortion manifold embeddings