RANSAC and its variants are widely used for robust estimation, however, they
commonly follow a greedy approach to finding the highest scoring model while
ignoring other model hypotheses. In contrast, Iteratively Reweighted Least
Squares (IRLS) techniques gradually approach the model by iteratively updating
the weight of each correspondence based on the residuals from previous
iterations. Inspired by these methods, we propose a new RANSAC framework that
learns to explore the parameter space by considering the residuals seen so far
via a novel attention layer. The attention mechanism operates on a batch of
point-to-model residuals, and updates a per-point estimation state to take into
account the consensus found through a lightweight one-step transformer. This
rich state then guides the minimal sampling between iterations as well as the
model refinement. We evaluate the proposed approach on essential and
fundamental matrix estimation on a number of indoor and outdoor datasets. It
outperforms state-of-the-art estimators by a significant margin adding only a
small runtime overhead. Moreover, we demonstrate good generalization properties
of our trained model, indicating its effectiveness across different datasets
and tasks. The proposed attention mechanism and one-step transformer provide an
adaptive behavior that enhances the performance of RANSAC, making it a more
effective tool for robust estimation. Code is available at
this https URL

通过引入注意机制和一步变压器，我们提出了一种新的 RANSAC 框架，其学习通过考虑迄今为止观察到的残差来探索参数空间，并在实验中表现出了显著的性能提升和良好的泛化性能。

共识自适应 RANSAC

Consensus-Adaptive RANSAC

We propose a new iteratively reweighted least squares (IRLS) algorithm for
the recovery of a matrix $X \in \mathbb{C}^{d_1\times d_2}$ of rank $r
\ll\min(d_1,d_2)$ from incomplete linear observations, solving a sequence of
low complexity linear problems. The easily implementable algorithm, which we
call harmonic mean iteratively reweighted least squares (HM-IRLS), optimizes a
non-convex Schatten-$p$ quasi-norm penalization to promote low-rankness and
carries three major strengths, in particular for the matrix completion setting.
First, we observe a remarkable global convergence behavior of the algorithm's
iterates to the low-rank matrix for relevant, interesting cases, for which any
other state-of-the-art optimization approach fails the recovery. Secondly,
HM-IRLS exhibits an empirical recovery probability close to $1$ even for a
number of measurements very close to the theoretical lower bound $r (d_1 +d_2
-r)$, i.e., already for significantly fewer linear observations than any other
tractable approach in the literature. Thirdly, HM-IRLS exhibits a locally
superlinear rate of convergence (of order $2-p$) if the linear observations
fulfill a suitable null space property. While for the first two properties we
have so far only strong empirical evidence, we prove the third property as our
main theoretical result.

本文提出了谐波均值迭代加权最小二乘（HM-IRLS）算法，应用于从不完整的线性观测中恢复秩为 r 的矩阵 X，通过一系列低复杂度的线性问题求解，以优化非凸的 Schatten-p 准范罚项，以提高低秩性。HM-IRLS 算法具有三个主要优势，尤其是在矩阵完成设置中：第一，算法自变量对于相关感兴趣的情况下以低秩矩阵呈现出显着的全局收敛性；第二，即使线性观察值的数量非常接近理论下界 r（d1+d2-r），HM-IRLS 表现出接近 1 的经验恢复概率；第三，如果线性观察满足适合的零空间属性，则 HM-IRLS 表现出局部超线性收敛速度（2-p）的优势。

迭代重新加权偏最小二乘调和平均用于低秩矩阵恢复

Harmonic Mean Iteratively Reweighted Least Squares for Low-Rank Matrix  Recovery

This work presents a general framework for solving the low rank and/or sparse
matrix minimization problems, which may involve multiple non-smooth terms. The
Iteratively Reweighted Least Squares (IRLS) method is a fast solver, which
smooths the objective function and minimizes it by alternately updating the
variables and their weights. However, the traditional IRLS can only solve a
sparse only or low rank only minimization problem with squared loss or an
affine constraint. This work generalizes IRLS to solve joint/mixed low rank and
sparse minimization problems, which are essential formulations for many tasks.
As a concrete example, we solve the Schatten-$p$ norm and $\ell_{2,q}$-norm
regularized Low-Rank Representation (LRR) problem by IRLS, and theoretically
prove that the derived solution is a stationary point (globally optimal if
$p,q\geq1$). Our convergence proof of IRLS is more general than previous one
which depends on the special properties of the Schatten-$p$ norm and
$\ell_{2,q}$-norm. Extensive experiments on both synthetic and real data sets
demonstrate that our IRLS is much more efficient.

本文提出了一个解决低秩和 / 或稀疏矩阵最小化问题的一般框架，使用迭代重新加权最小二乘（IRLS）方法来解决混合低秩和稀疏最小化问题，例如用于解决 Schatten-p 规范和 ell_2,q-norm 规范的低秩表示问题，理论证明了所获得的解为静止点，并在合成和实际数据集上进行了广泛的实验以证明其有效性。

迭代加权最小二乘法平滑低秩和稀疏矩阵恢复

Smoothed Low Rank and Sparse Matrix Recovery by Iteratively Reweighted  Least Squares Minimization

We present and analyze an efficient implementation of an iteratively
reweighted least squares algorithm for recovering a matrix from a small number
of linear measurements. The algorithm is designed for the simultaneous
promotion of both a minimal nuclear norm and an approximatively low-rank
solution. Under the assumption that the linear measurements fulfill a suitable
generalization of the Null Space Property known in the context of compressed
sensing, the algorithm is guaranteed to recover iteratively any matrix with an
error of the order of the best k-rank approximation. In certain relevant cases,
for instance for the matrix completion problem, our version of this algorithm
can take advantage of the Woodbury matrix identity, which allows to expedite
the solution of the least squares problems required at each iteration. We
present numerical experiments that confirm the robustness of the algorithm for
the solution of matrix completion problems, and demonstrate its competitiveness
with respect to other techniques proposed recently in the literature.

使用迭代重加权最小二乘算法，同时结合核范数和近似低秩解的最小化，有效地从少量线性测量中恢复矩阵，并通过实验证明在矩阵完成问题中具有竞争力。