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  1. We present a computationally-efficient and numerically-robust algorithm for finding real roots of polynomials. It begins with determining the intervals where the given polynomial is monotonic. Then, it performs a robust variant of Newton iterations to find the real root within each interval, providing fast and guaranteed convergence and satisfying the given error bound, as permitted by the numerical precision used. For cubic polynomials, the algorithm is more accurate and faster than both the analytical solution and directly applying Newton iterations. It trivially extends to polynomials with arbitrary degrees, but it is limited to finding the real roots only and has quadratic worst-case complexity in terms of the polynomial's degree. We show that our method outperforms alternative polynomial solutions we tested up to degree 20. We also present an example rendering application with a known efficient numerical solution and show that our method provides faster, more accurate, and more robust solutions by solving polynomials of degree 10.
    Free, publicly-accessible full text available July 25, 2023
  2. Neural networks tend to achieve better accuracy with training if they are larger — even if the resulting models are overparameterized. Nevertheless, carefully removing such excess of parameters before, during, or after training may also produce models with similar or even improved accuracy. In many cases, that can be curiously achieved by heuristics as simple as removing a percentage of the weights with the smallest absolute value — even though absolute value is not a perfect proxy for weight relevance. With the premise that obtaining significantly better performance from pruning depends on accounting for the combined effect of removing multiple weights, we revisit one of the classic approaches for impact-based pruning: the Optimal Brain Surgeon (OBS). We propose a tractable heuristic for solving the combinatorial extension of OBS, in which we select weights for simultaneous removal, and we combine it with a single-pass systematic update of unpruned weights. Our selection method outperforms other methods for high sparsity, and the single-pass weight update is also advantageous if applied after those methods.
  3. The ability to estimate the 3D human shape and pose from images can be useful in many contexts. Recent approaches have explored using graph convolutional networks and achieved promising results. The fact that the 3D shape is represented by a mesh, an undirected graph, makes graph convolutional networks a natural fit for this problem. However, graph convolutional networks have limited representation power Information from nodes in the graph is passed to connected neighbors, and propagation of information requires successive graph convolutions. To overcome this limitation, we propose a dual-scale graph approach. We use a coarse graph, derived from a dense graph, to estimate the human’s 3D pose, and the dense graph to estimate the 3D shape. Information in coarse graphs can be propagated over longer distances compared to dense graphs. In addition, information about pose can guide to recover local shape detail and vice versa. We recognize that the connection between coarse and dense is itself a graph, and introduce graph fusion blocks to exchange information between graphs with different scales. We train our model end-to-end and show that we can achieve state-of-the-art results for several evaluation datasets. The code is available at the following link,
  4. We can compress a rectifier network while exactly preserving its underlying functionality with respect to a given input domain if some of its neurons are stable. However, current approaches to determine the stability of neurons with Rectified Linear Unit (ReLU) activations require solving or finding a good approximation to multiple discrete optimization problems. In this work, we introduce an algorithm based on solving a single optimization problem to identify all stable neurons. Our approach is on median 183 times faster than the state-of-art method on CIFAR-10, which allows us to explore exact compression on deeper (5 x 100) and wider (2 x 800) networks within minutes. For classifiers trained under an amount of L1 regularization that does not worsen accuracy, we can remove up to 56% of the connections on the CIFAR-10 dataset. The code is available at the following link,
  5. Ishikawa, H. ; Liu, CL. ; Pajdla, T. ; Shi, J. (Ed.)
    We propose a novel technique to register sparse 3D scans in the absence of texture. While existing methods such as KinectFusion or Iterative Closest Points (ICP) heavily rely on dense point clouds, this task is particularly challenging under sparse conditions without RGB data. Sparse texture-less data does not come with high-quality boundary signal, and this prohibits the use of correspondences from corners, junctions, or boundary lines. Moreover, in the case of sparse data, it is incorrect to assume that the same point will be captured in two consecutive scans. We take a different approach and first re-parameterize the point-cloud using a large number of line segments. In this re-parameterized data, there exists a large number of line intersection (and not correspondence) constraints that allow us to solve the registration task. We propose the use of a two-step alternating projection algorithm by formulating the registration as the simultaneous satisfaction of intersection and rigidity constraints. The proposed approach outperforms other top-scoring algorithms on both Kinect and LiDAR datasets. In Kinect, we can use 100X downsampled sparse data and still outperform competing methods operating on full-resolution data.
  6. 3D scan registration is a classical, yet a highly useful problem in the context of 3D sensors such as Kinect and Velodyne. While there are several existing methods, the techniques are usually incremental where adjacent scans are registered first to obtain the initial poses, followed by motion averaging and bundle-adjustment refinement. In this paper, we take a different approach and develop minimal solvers for jointly computing the initial poses of cameras in small loops such as 3-, 4-, and 5-cycles1. Note that the classical registration of 2 scans can be done using a minimum of 3 point matches to compute 6 degrees of relative motion. On the other hand, to jointly compute the 3D reg- istrations in n-cycles, we take 2 point matches between the first n−1 consecutive pairs (i.e., Scan 1 & Scan 2, . . . , and Scan n − 1 & Scan n) and 1 or 2 point matches between Scan 1 and Scan n. Overall, we use 5, 7, and 10 point matches for 3-, 4-, and 5-cycles, and recover 12, 18, and 24 degrees of transformation variables, respectively. Using simulations and real-data we show that the 3D registration using mini n-cycles are computationally efficient,more »and can provide alternate and better initial poses compared to standard pairwise methods.« less