We consider the problem of linear estimation, and establish an extension of
the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero
but bounded with respect to a matrix norm of Schatten type. We derive simple
and explicit formulas for the optimal estimator in the cases of Nuclear and
Spectral norms (with the Frobenius case recovering ridge regression).
Additionally, we analytically derive the generalization error in multiple
random matrix ensembles, and compare with Ridge regression. Finally, we conduct
an extensive simulation study, in which we show that the cross-validated
Nuclear and Spectral regressors can outperform Ridge in several circumstances.

线性估计问题中，我们建立了一个高斯 - 马尔可夫定理的扩展，其中偏置算子允许非零但相对于 Schatten 类型的矩阵范数保持有界。我们导出了核范数和谱范数（弗罗贝尼乌斯范数情况下回收岭回归）的最优估计器的简单和明确公式。此外，我们在多个随机矩阵集合中分析地导出了广义误差并与岭回归进行比较。最后，我们进行了大量模拟研究，结果显示交叉验证的核范数和谱范数回归器在某些情况下能胜过岭回归。

线性估计中的平坦极小值和扩展的高斯马尔可夫定理

Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem

This paper proposes a new extrinsic calibration of kaleidoscopic imaging
system by estimating normals and distances of the mirrors. The problem to be
solved in this paper is a simultaneous estimation of all mirror parameters
consistent throughout multiple reflections. Unlike conventional methods
utilizing a pair of direct and mirrored images of a reference 3D object to
estimate the parameters on a per-mirror basis, our method renders the
simultaneous estimation problem into solving a linear set of equations. The key
contribution of this paper is to introduce a linear estimation of multiple
mirror parameters from kaleidoscopic 2D projections of a single 3D point of
unknown geometry. Evaluations with synthesized and real images demonstrate the
performance of the proposed algorithm in comparison with conventional methods.

本论文提出了一种通过估计镜面的法线和距离的新型复合成像系统的外部校准方法。与传统方法不同，该论文的解决方案是同时估计所有镜参数，该方法将同时估计问题转化为解决一组线性方程，具体表现为从单个未知几何形状的三维点的万花筒二维投影中线性估计多个镜参数，通过合成和实际图像的评价显示了该算法与传统方法的性能对比。

基于单个三维点的万花筒成像系统线性外部标定

A Linear Extrinsic Calibration of Kaleidoscopic Imaging System from  Single 3D Point

Consider measuring an n-dimensional vector x through the inner product with
several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal
processing and statistics to assume the linear response model y_i = <a_i, x> +
e_i, where e_i is a noise term. However, in practice the precise relationship
between the signal x and the observations y_i may not follow the linear model,
and in some cases it may not even be known. To address this challenge, in this
paper we propose a general model where it is only assumed that each observation
y_i may depend on a_i only through <a_i, x>. We do not assume that the
dependence is known. This is a form of the semiparametric single index model,
and it includes the linear model as well as many forms of the generalized
linear model as special cases. We further assume that the signal x has some
structure, and we formulate this as a general assumption that x belongs to some
known (but arbitrary) feasible set K. We carefully detail the benefit of using
the signal structure to improve estimation. The theory is based on the mean
width of K, a geometric parameter which can be used to understand its effective
dimension in estimation problems. We determine a simple, efficient two-step
procedure for estimating the signal based on this model -- a linear estimation
followed by metric projection onto K. We give general conditions under which
the estimator is minimax optimal up to a constant. This leads to the intriguing
conclusion that in the high noise regime, an unknown non-linearity in the
observations does not significantly reduce one's ability to determine the
signal, even when the non-linearity may be non-invertible. Our results may be
specialized to understand the effect of non-linearities in compressed sensing.

本文提出了一种简化的半参数单指数模型，用于信号处理中的估计问题，理论基于可行集的平均宽度并通过线性估计和度量投影实现，即使在高噪声情形下，未知的非线性关系也不会显著降低确定信号的能力。