arXiv research

Rotation invariant algorithms fail with hard labels sampled from sparse targets.

The scalar curvature equation for rotation invariant Kähler metrics on $\mathbb{C}^n \backslash \{0\}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $\mathbb{C}^n \backslash \{0\}$ in lower dimensions. We also prove that there doe…

We classify Kähler-Einstein manifolds which admit a Kähler immersion into a finite dimensional complex projective space endowed with the Fubini-Study metric, whose codimention is not greater than 3 and whose metric is rotation invariant.

Rotation invariant algorithms fail on sparse problems even with noise.

GCNNs gain rotation invariance with more training augmentation, making SVD-Universal more effective.

High-dimensional kernel regression struggles due to rotational invariance.

Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal function approximation in an arbitrarily high dimensional Euclidian space. It is show…

Recent years have witnessed the emergence and increasing popularity of 3D medical imaging techniques with the development of 3D sensors and technology. However, achieving geometric invariance in the processing of 3D medical images is computationally expensive but nonetheless essential due to the presence of possible er…

Solves classical problem with Kähler-Einstein metrics in complex projective spaces.

We introduce a novel class of rotation invariants of two dimensional curves based on iterated integrals. The invariants we present are in some sense complete and we describe an algorithm to calculate them, giving explicit computations up to order six. We present an application to online (stroke-trajectory based) charac…

New MCMC method improves sampling efficiency across diverse structural models.

This paper presents a novel approach to exploit the distinctive invariant features in convolutional neural network. The proposed CNN model uses Scale Invariant Feature Transform (SIFT) descriptor instead of the max-pooling layer. Max-pooling layer discards the pose, i.e., translational and rotational relationship betwe…

A new sliced IGW distance for Gromov-Wasserstein alignment.

We investigate the problem of estimating a given real symmetric signal matrix $\textbf{C}$ from a noisy observation matrix $\textbf{M}$ in the limit of large dimension. We consider the case where the noisy measurement $\textbf{M}$ comes either from an arbitrary additive or multiplicative rotational invariant perturbati…

In recent years, convolutional neural networks (CNN) have played an important role in the field of deep learning. Variants of CNN's have proven to be very successful in classification tasks across different domains. However, there are two big drawbacks to CNN's: their failure to take into account of important spatial h…

New method learns disentangled discrete representations using categorical variational autoencoders.

The paper develops a new algorithm for RBMs using dynamical mean-field theory.

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…

We consider probabilistic PCA and related factor models from a Bayesian perspective. These models are in general not identifiable as the likelihood has a rotational symmetry. This gives rise to complicated posterior distributions with continuous subspaces of equal density and thus hinders efficiency of inference as wel…

We prove two results on the classification of trivial Legendrian embeddings $g: G \rightarrow (S^3,ξ_{std})$ of planar graphs. First, the oriented Legendrian ribbon $R_g$ and rotation invariant $\text{rot}_g$ are a complete set of invariants. Second, if $G$ is 3-connected or contains $K_4$ as a minor, then the unique t…

A new Wasserstein distance method for comparing incomparable distributions.

The paper develops GPR models for hyperelastic materials, improving accuracy and rotational invariance.

A novel approach learns goal-conditioned policies for locomotion using batch RL.

Study connects covariance cleaning theory to information theory for heavy-tailed distributions.

We propose a principled method for kernel learning, which relies on a Fourier-analytic characterization of translation-invariant or rotation-invariant kernels. Our method produces a sequence of feature maps, iteratively refining the SVM margin. We provide rigorous guarantees for optimality and generalization, interpret…

A new method calculates intrinsic effective sample size for manifold-valued data.

New method simplifies tomographic reconstruction using RKHS.

Spherical data is found in many applications. By modeling the discretized sphere as a graph, we can accommodate non-uniformly distributed, partial, and changing samplings. Moreover, graph convolutions are computationally more efficient than spherical convolutions. As equivariance is desired to exploit rotational symmet…

A new method for density estimation using mixture discrepancy and moments.

We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the …

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

Bayes-optimal limits in PCA with structured noise are determined.

A machine learning model with approximate rotational symmetry is tested and found stable.

Performance of neural networks can be significantly improved by encoding known invariance for particular tasks. Many image classification tasks, such as those related to cellular imaging, exhibit invariance to rotation. We present a novel scheme using the magnitude response of the 2D-discrete-Fourier transform (2D-DFT)…

All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $ξ$. We generalize…

Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but…

We add prior knowledge to deep networks to make them invariant to transformations.

We introduce a novel co-learning paradigm for manifolds naturally equipped with a group action, motivated by recent developments on learning a manifold from attached fibre bundle structures. We utilize a representation theoretic mechanism that canonically associates multiple independent vector bundles over a common bas…

Let $M\subset{\complex}^n$ be a complex domain of ${\complex}^n$ endowed with a rotation invariant \K form $ω_Φ= \frac{i}{2} \partial\bar\partialΦ$. In this paper we describe sufficient conditions on the \K potential $Φ$ for $(M, ω_Φ)$ to admit a symplectic embedding (explicitely described in terms of $Φ$) into a compl…

Recent work (Cohen & Welling, 2016) has shown that generalizations of convolutions, based on group theory, provide powerful inductive biases for learning. In these generalizations, filters are not only translated but can also be rotated, flipped, etc. However, coming up with exact models of how to rotate a 3 x 3 filter…

This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complex…

Fetal brain imaging is a cornerstone of prenatal screening and early diagnosis of congenital anomalies. Knowledge of fetal gestational age is the key to the accurate assessment of brain development. This study develops an attention-based deep learning model to predict gestational age of the fetal brain. The proposed mo…

Spectral clustering performance depends on eigenvector fluctuations, shown to be Gaussian.

Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example based on principal com…

Study analyzes Bayesian inference algorithms using dynamical functional approach.

New weighted surface area measures for convex bodies with applications.

New theory allows ICA without assuming non-Gaussian sources.

Strong inductive biases prevent harmless interpolation in overparameterized models.