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1. Fast Zernike fitting of freeform surfaces using the Gauss-Legendre quadrature

   https://doi.org/10.1364/OE.523022  

   Fan, Yiwen; Forbes, G_W; Rolland, Jannick_P (May 2024, Optics Express)

   Zernike polynomial orthogonality, an established mathematical principle, is leveraged with the Gauss-Legendre quadrature rule in a rapid novel approach to fitting data over a circular domain. This approach provides significantly faster fitting speeds, in the order of thousands of times, while maintaining comparable error rates achieved with conventional least-square fitting techniques. We demonstrate the technique for fitting mid-spatial-frequencies (MSF) prevalent in small-tool-manufacturing typical of aspheric and freeform optics that are poised to soon permeate a wide range of optical technologies. 

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2. Kernel Truncated Randomized Ridge Regression: Optimal Rates and Low Noise Acceleration

   Jun, Kwang-Sung; Cutkosky, Ashok; Orabona, Francesco (January 2019, Advances in neural information processing systems)

   Wallach, H.; Larochelle, H.; Beygelzimer, A.; d'Alché-Buc, F.; null; Garnett, R. (Ed.)

   In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated integral operator and the complexity of the optimal predictor measured through the integral operator. 

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3. Elastic Shape Analysis of Tree-Like 3D Objects Using Extended SRVF Representation

   https://doi.org/10.1109/TPAMI.2023.3334525  

   Wang, Guan; Laga, Hamid; Srivastava, Anuj (April 2024, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   How can one analyze detailed 3D biological objects, such as neuronal and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects - each subtree has a main branch with some side branches attached - and one needs to match these structures across objects for meaningful comparisons. We propose a novel representation that extends the Square-Root Velocity Function (SRVF), initially developed for Euclidean curves, to tree-shaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one tree-shaped object into the other. Compared to the current metrics such as the Quotient Euclidean Distance (QED) and the Tree Edit Distance (TED), the proposed representation and metric capture the full elasticity of the branches (i.e., bending and stretching) as well as the topological variations (i.e., branch death/birth and sliding). It completely avoids the shrinkage that results from the edge collapse and node split operations of the QED and TED metrics. We demonstrate the utility of this framework in comparing, matching, and computing geodesics between biological objects such as neuronal and botanical trees. We also demonstrate its application to various shape analysis tasks such as (i) symmetry analysis and symmetrization of tree-shaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of tree-shaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel tree-shaped 3D objects through random sampling from estimated probability distributions. 

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4. Time/Accuracy Tradeoffs for learning a ReLU with respect to Gaussian Marginals

   Goel, Surbhi; Karmalkar, Sushrut; Klivans, Adam (January 2019, Advances in neural information processing systems)

   We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let 𝗈𝗉𝗍<1 be the population loss of the best-fitting ReLU. We prove: 1. Finding a ReLU with square-loss 𝗈𝗉𝗍+ϵ is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time. 2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error O(𝗈𝗉𝗍^{2/3}). The algorithm uses a novel reduction to noisy halfspace learning with respect to 0/1 loss. Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU. 

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5. Cross-Modal Center Loss for 3D Cross-Modal Retrieval

   Jing, L; Vahdani, E; Tan, J; Tian, Y. (June 2021, IEEE Conference on Computer Vision and Pattern Recognition (CVPR))

   null (Ed.)

   Cross-modal retrieval aims to learn discriminative and modal-invariant features for data from different modalities. Unlike the existing methods which usually learn from the features extracted by offline networks, in this paper, we pro- pose an approach to jointly train the components of cross- modal retrieval framework with metadata, and enable the network to find optimal features. The proposed end-to-end framework is updated with three loss functions: 1) a novel cross-modal center loss to eliminate cross-modal discrepancy, 2) cross-entropy loss to maximize inter-class variations, and 3) mean-square-error loss to reduce modality variations. In particular, our proposed cross-modal center loss minimizes the distances of features from objects belonging to the same class across all modalities. Extensive experiments have been conducted on the retrieval tasks across multi-modalities including 2D image, 3D point cloud and mesh data. The proposed framework significantly outperforms the state-of-the-art methods for both cross-modal and in-domain retrieval for 3D objects on the ModelNet10 and ModelNet40 datasets. 

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