Learning structured representations of the visual world in terms of objects
promises to significantly improve the generalization abilities of current
machine learning models. While recent efforts to this end have shown promising
empirical progress, a theoretical account of when unsupervised object-centric
representation learning is possible is still lacking. Consequently,
understanding the reasons for the success of existing object-centric methods as
well as designing new theoretically grounded methods remains challenging. In
the present work, we analyze when object-centric representations can provably
be learned without supervision. To this end, we first introduce two assumptions
on the generative process for scenes comprised of several objects, which we
call compositionality and irreducibility. Under this generative process, we
prove that the ground-truth object representations can be identified by an
invertible and compositional inference model, even in the presence of
dependencies between objects. We empirically validate our results through
experiments on synthetic data. Finally, we provide evidence that our theory
holds predictive power for existing object-centric models by showing a close
correspondence between models' compositionality and invertibility and their
empirical identifiability.

通过引入组合性和不可约性假设，分析了何时可以在无监督情况下学习对象为中心的表示，并通过在合成数据上的实验验证了结果。还提供了证据表明该理论具有现有对象为中心模型的预测能力。

可证明学习物体为中心的表达

Provably Learning Object-Centric Representations

This paper discusses the irreducibility and geometric ergodicity of the
Hamiltonian Monte Carlo (HMC) algorithm. We consider cases where the number of
steps of the symplectic integrator is either fixed or random. Under mild
conditions on the potential $\F$ associated with target distribution $\pi$, we
first show that the Markov kernel associated to the HMC algorithm is
irreducible and recurrent. Under more stringent conditions, we then establish
that the Markov kernel is Harris recurrent. Finally, we provide verifiable
conditions on $\F$ under which the HMC sampler is geometrically ergodic.

本文讨论了 Hamiltonian Monte Carlo 算法的不可约性和几何遍历性，考虑了较为宽松和严格的条件，得出了 HMC 采样器具有几何遍历性的可验证条件。

Hamiltonian Monte Carlo 收敛性研究

On the convergence of Hamiltonian Monte Carlo

We find geometric and arithmetic conditions in order to characterize the
irreducibility of the determinant of the generic Vandermonde matrix over the
algebraic closure of any field k. We also characterize those determinants whose
tropicalization with respect to the variables of a row is irreducible.

本文通过找到几何和算术条件，对在任意域 k 的代数闭包上一般范德蒙德矩阵的行列式的可约化进行了特征化，并确定了那些在某一行的变量的张量积依旧可约的行列式。