23 research outputs found

    Robust Object Classification Approach using Spherical Harmonics

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    Point clouds produced by either 3D scanners or multi-view images are often imperfect and contain noise or outliers. This paper presents an end-to-end robust spherical harmonics approach to classifying 3D objects. The proposed framework first uses the voxel grid of concentric spheres to learn features over the unit ball. We then limit the spherical harmonics order level to suppress the effect of noise and outliers. In addition, the entire classification operation is performed in the Fourier domain. As a result, our proposed model learned features that are less sensitive to data perturbations and corruptions. We tested our proposed model against several types of data perturbations and corruptions, such as noise and outliers. Our results show that the proposed model has fewer parameters, competes with state-of-art networks in terms of robustness to data inaccuracies, and is faster than other robust methods. Our implementation code is also publicly available1

    On incorporating inductive biases into deep neural networks

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    A machine learning (ML) algorithm can be interpreted as a system that learns to capture patterns in data distributions. Before the modern \emph{deep learning era}, emulating the human brain, the use of structured representations and strong inductive bias have been prevalent in building ML models, partly due to the expensive computational resources and the limited availability of data. On the contrary, armed with increasingly cheaper hardware and abundant data, deep learning has made unprecedented progress during the past decade, showcasing incredible performance on a diverse set of ML tasks. In contrast to \emph{classical ML} models, the latter seeks to minimize structured representations and inductive bias when learning, implicitly favoring the flexibility of learning over manual intervention. Despite the impressive performance, attention is being drawn towards enhancing the (relatively) weaker areas of deep models such as learning with limited resources, robustness, minimal overhead to realize simple relationships, and ability to generalize the learned representations beyond the training conditions, which were (arguably) the forte of classical ML. Consequently, a recent hybrid trend is surfacing that aims to blend structured representations and substantial inductive bias into deep models, with the hope of improving them. Based on the above motivation, this thesis investigates methods to improve the performance of deep models using inductive bias and structured representations across multiple problem domains. To this end, we inject a priori knowledge into deep models in the form of enhanced feature extraction techniques, geometrical priors, engineered features, and optimization constraints. Especially, we show that by leveraging the prior knowledge about the task in hand and the structure of data, the performance of deep learning models can be significantly elevated. We begin by exploring equivariant representation learning. In general, the real-world observations are prone to fundamental transformations (e.g., translation, rotation), and deep models typically demand expensive data-augmentations and a high number of filters to tackle such variance. In comparison, carefully designed equivariant filters possess this ability by nature. Henceforth, we propose a novel \emph{volumetric convolution} operation that can convolve arbitrary functions in the unit-ball (B3\mathbb{B}^3) while preserving rotational equivariance by projecting the input data onto the Zernike basis. We conduct extensive experiments and show that our formulations can be used to construct significantly cheaper ML models. Next, we study generative modeling of 3D objects and propose a principled approach to synthesize 3D point-clouds in the spectral-domain by obtaining a structured representation of 3D points as functions on the unit sphere (S2\mathbb{S}^2). Using the prior knowledge about the spectral moments and the output data manifold, we design an architecture that can maximally utilize the information in the inputs and generate high-resolution point-clouds with minimal computational overhead. Finally, we propose a framework to build normalizing flows (NF) based on increasing triangular maps and Bernstein-type polynomials. Compared to the existing NF approaches, our framework consists of favorable characteristics for fusing inductive bias within the model i.e., theoretical upper bounds for the approximation error, robustness, higher interpretability, suitability for compactly supported densities, and the ability to employ higher degree polynomials without training instability. Most importantly, we present a constructive universality proof, which permits us to analytically derive the optimal model coefficients for known transformations without training

    Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall

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    Computational fluid dynamics (CFD) is a valuable asset for patient-specific cardiovascular-disease diagnosis and prognosis, but its high computational demands hamper its adoption in practice. Machine-learning methods that estimate blood flow in individual patients could accelerate or replace CFD simulation to overcome these limitations. In this work, we consider the estimation of vector-valued quantities on the wall of three-dimensional geometric artery models. We employ group-equivariant graph convolution in an end-to-end SE(3)-equivariant neural network that operates directly on triangular surface meshes and makes efficient use of training data. We run experiments on a large dataset of synthetic coronary arteries and find that our method estimates directional wall shear stress (WSS) with an approximation error of 7.6% and normalised mean absolute error (NMAE) of 0.4% while up to two orders of magnitude faster than CFD. Furthermore, we show that our method is powerful enough to accurately predict transient, vector-valued WSS over the cardiac cycle while conditioned on a range of different inflow boundary conditions. These results demonstrate the potential of our proposed method as a plugin replacement for CFD in the personalised prediction of hemodynamic vector and scalar fields.Comment: Preprint. Under Revie

    Geometric deep learning and equivariant neural networks

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    We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning
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