Huge progress has been made in LiDAR 3D semantic segmentation, but there are two under-explored imbalances on the radial axis: points are unevenly concentrated on the near side, and the distribution of foreground object instances is skewed to the near side. This leads the training of the model to favor semantics at the near side with the majority of points and object instances. Both the cylindrical and the spherical voxelizations aim to address the problem of imbalanced point distribution by increasing the volume of voxels along the radial distance to include fewer near-side points in a smaller voxel and more far-side points in a bigger voxel. However, this causes a problem of the receptive field enlarging along the radial distance, which is not desirable in LiDAR 3D segmentation. This can be addressed in cubic voxelization which has a fixed volume of voxels. Thus, we propose a new LiDAR 3D semantic segmentation network (Hi-VoxelNet) with Hybrid Voxelization that leverages the advantages of cubic, cylindrical, and spherical voxelizations for hybrid voxel feature learning. To address the radial imbalance of object instances, we propose a novel data augmentation technique termed as RadialMix that uses radial sample duplication to increase the number of distant foreground object instances and mixes the radial duplication with another point cloud for enriching the training samples. With the joint improvements of the radial imbalances, our method archives state-of-the-art performance on nuScenes and SemanticKITTI datasets, and it shows significant improvements along the radial axis. Our code is publicly available at https://github.com/jialeli1/lidarseg3d.

Covariance estimation for high dimensional signals is a classically difficult problem in statistical signal analysis and machine learning. In this paper, we propose a maximum likelihood (ML) approach to covariance estimation, which employs a novel non-linear sparsity constraint. More specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy optimization of the log-likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The resulting estimator is generally positive definite and well-conditioned, even when the sample size is limited. Experiments on a combination of simulated data, standard hyperspectral data, and face image sets show that the SMT-based covariance estimates are consistently more accurate than both traditional shrinkage estimates and recently proposed graphical lasso estimates for a variety of different classes and sample sizes. An important property of the new covariance estimate is that it naturally yields a fast implementation of the estimated eigen-transformation using the SMT representation. In fact, the SMT can be viewed as a generalization of the classical fast Fourier transform (FFT) in that it uses "butterflies" to represent an orthonormal transform. However, unlike the FFT, the SMT can be used for fast eigen-signal analysis of general non-stationary signals.

Recently, the Sparse Matrix Transform (SMT) has been proposed as a tool for estimating the eigen-decomposition of high dimensional data vectors. The SMT approach has two major advantages: First it can improve the accuracy of the eigendecomposition, particularly when the number of observations, n, is less the the vector dimension, p. Second, the resulting SMT eigen-decomposition is very fast to apply, i.e. O(p). In this paper, we present an SMT eigen-decomposition method suited for application to signals that live on graphs. This new SMT eigen-decomposition method has two major advantages over the more generic method presented in. First, the resulting SMT can be more accurately estimated due to the graphical constraint. Second, the computation required to design the SMT from training data is dramatically reduced from an average observed complexity of p3to p log p.

Regression from high dimensional observation vectors is particularly difficult when training data is limited. More specifically, if the number of sample vectors n is less than dimension of the sample vectors p, then accurate regression is difficult to perform without prior knowledge of the data covariance. In this paper, we propose a novel approach to high dimensional regression for application when n ≪ p. The approach works by first decorrelating the high dimensional observation vector using the sparse matrix transform (SMT) estimate of the data covariance. Then the decorrelated observations are used in a regularized regression procedure such as Lasso or shrinkage. Numerical results demonstrate that the proposed regression approach can significantly improve the prediction accuracy, especially when n is small and the signal to be predicted lies in the subspace of the observations corresponding to the small eigenvalues.

We investigate three algorithms that use the sparse matrix transform (SMT) to produce variance-maximizing linear projections to a lower-dimensional space. The SMT expresses the projection as a sequence of Givens rotations and this enables computationally efficient implementation of the projection operator. The baseline algorithm uses the SMT to directly approximate the optimal solution that is given by principal components analysis (PCA). A variant of the baseline begins with a standard SMT solution, but prunes the sequence of Givens rotations to only include those that contribute to the variance maximization. Finally, a simpler and faster third algorithm is introduced; this also estimates the projection operator with a sequence of Givens rotations, but in this case, the rotations are chosen to optimize a criterion that more directly expresses the dimension reduction criterion.

We present a method for noniterative maximum a posteriori (MAP) tomographic reconstruction which is based on the use of sparse matrix representations. Our approach is to precompute and store the inverse matrix required for MAP reconstruction. This approach has generally not been used in the past because the inverse matrix is typically large and fully populated (i.e., not sparse). In order to overcome this problem, we introduce two new ideas. The first idea is a novel theory for the lossy source coding of matrix transformations which we refer to as matrix source coding. This theory is based on a distortion metric that reflects the distortions produced in the final matrix-vector product, rather than the distortions in the coded matrix itself. The resulting algorithms are shown to require orthonormal transformations of both the measurement data and the matrix rows and columns before quantization and coding. The second idea is a method for efficiently storing and computing the required orthonormal transformations, which we call a sparse-matrix transform (SMT). The SMT is a generalization of the classical FFT in that it uses butterflies to compute an orthonormal transform; but unlike an FFT, the SMT uses the butterflies in an irregular pattern, and is numerically designed to best approximate the desired transforms. We demonstrate the potential of the noniterative MAP reconstruction with examples from optical tomography. The method requires offline computation to encode the inverse transform. However, once these offline computations are completed, the noniterative MAP algorithm is shown to reduce both storage and computation by well over two orders of magnitude, as compared to a linear iterative reconstruction methods.

Covariance estimation for high dimensional vectors is a classically difficult problem in statistical analysis and machine learning due to limited sample size. In this paper, we propose a new approach to covariance estimation, which is based on constrained maximum likelihood (ML) estimation of the covariance. Specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The estimator obtained using this method is always positive definite and well-conditioned even with limited sample size. Experiments on hyperspectral data show that SMT covariance estimation results in consistently better estimates of the covariance for a variety of different classes and sample sizes compared to traditional shrinkage estimators.