In this work, we propose a novel dimensionality reduction technique, DiffRed,
which first projects the data matrix, A, along first $k_1$ principal components
and the residual matrix $A^{*}$ (left after subtracting its $k_1$-rank
approximation) along $k_2$ Gaussian random vectors. We evaluate M1, the
distortion of mean-squared pair-wise distance, and Stress, the normalized value
of RMS of distortion of the pairwise distances. We rigorously prove that
DiffRed achieves a general upper bound of
$O\left(\sqrt{\frac{1-p}{k_2}}\right)$ on Stress and
$O\left(\frac{(1-p)}{\sqrt{k_2*\rho(A^{*})}}\right)$ on M1 where $p$ is the
fraction of variance explained by the first $k_1$ principal components and
$\rho(A^{*})$ is the stable rank of $A^{*}$. These bounds are tighter than the
currently known results for Random maps. Our extensive experiments on a variety
of real-world datasets demonstrate that DiffRed achieves near zero M1 and much
lower values of Stress as compared to the well-known dimensionality reduction
techniques. In particular, DiffRed can map a 6 million dimensional dataset to
10 dimensions with 54% lower Stress than PCA.

通过使用 DiffRed 技术，我们可以将多维数据集映射到低维空间，并获得比常用降维技术更低的扭曲度和应力值。

DiffRed：稳定秩引导的降维

DiffRed: Dimensionality Reduction guided by stable rank

We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian
random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as
$n, d \to \infty$. This problem is conjectured to have a sharp feasibility
transition: for any $\varepsilon > 0$, if $n \leq (1 - \varepsilon) d^2 / 4$
then $(\mathrm{P})$ has a solution with high probability, while $(\mathrm{P})$
has no solutions with high probability if $n \geq (1 + \varepsilon) d^2 /4$. So
far, only a trivial bound $n \geq d^2 / 2$ is known on the negative side, while
the best results on the positive side assume $n \leq d^2 /
\mathrm{polylog}(d)$. In this work, we improve over previous approaches using a
key result of Bartl & Mendelson on the concentration of Gram matrices of random
vectors under mild assumptions on their tail behavior. This allows us to give a
simple proof that $(\mathrm{P})$ is feasible with high probability when $n \leq
d^2 / C$, for a (possibly large) constant $C > 0$.

研究拟合 n 个高斯随机向量到以原点为中心的椭球体边界问题，证明了它在 n 与 d 趋近于无穷大的情况下具有尖锐的可行性转变，并使用 Bartl&Mendelson 的关键结果和矩阵专业性质得出了当 n ≤ d²/C 时 (此处 C>0 是一个常数)，该问题是可行的且概率高。

对二次随机点拟合椭球

Fitting an ellipsoid to a quadratic number of random points

We present a method for estimating conditionally Gaussian random vectors with
random covariance matrices, which uses techniques from the field of machine
learning. Such models are typical in communication systems, where the
covariance matrix of the channel vector depends on random parameters, e.g.,
angles of propagation paths. If the covariance matrices exhibit certain
Toeplitz and shift-invariance structures, the complexity of the MMSE channel
estimator can be reduced to O(M log M) floating point operations, where M is
the channel dimension. While in the absence of structure the complexity is much
higher, we obtain a similarly efficient (but suboptimal) estimator by using the
MMSE estimator of the structured model as a blueprint for the architecture of a
neural network. This network learns the MMSE estimator for the unstructured
model, but only within the given class of estimators that contains the MMSE
estimator for the structured model. Numerical simulations with typical spatial
channel models demonstrate the generalization properties of the chosen class of
estimators to realistic channel models.

本文介绍了一种使用机器学习技术来估计具有随机协方差矩阵的条件高斯随机向量的方法。这种模型在通信系统中非常常见，通过使用建立在结构化模型上 MMSE 估计器的神经网络蓝图，可以在给定精度下获得相似有效的解决方案，同时具备对理论模型之外的实际环境的一些基础的适应性。

学习 MMSE 信道估计器

Learning the MMSE Channel Estimator

In one-bit compressed sensing, previous results state that sparse signals may
be robustly recovered when the measurements are taken using Gaussian random
vectors. In contrast to standard compressed sensing, these results are not
extendable to natural non-Gaussian distributions without further assumptions,
as can be demonstrated by simple counter-examples. We show that approximately
sparse signals that are not extremely sparse can be accurately reconstructed
from single-bit measurements sampled according to a sub-gaussian distribution,
and the reconstruction comes as the solution to a convex program.

本研究通过使用次高斯分布的单比特测量，结合凸程序解决方案，可以精确地重建几乎稀疏的信号。