Given finite metric spaces $(X, d_X)$ and $(Y, d_Y)$, we investigate the
persistent homology $PH_*(X \times Y)$ of the Cartesian product $X \times Y$
equipped with the sum metric $d_X + d_Y$. Interpreting persistent homology as a
module over a polynomial ring, one might expect the usual K\"unneth short exact
sequence to hold. We prove that it holds for $PH_0$ and $PH_1$, and we
illustrate with the Hamming cube $\{0,1\}^k$ that it fails for $PH_n,\,\, n
\geq 2$. For $n = 2$, the prediction for $PH_2(X \times Y)$ from the expected
K\"unneth short exact sequence has a natural surjection onto $PH_2(X \times
Y)$. We compute the nontrivial kernel of this surjection for the splitting of
Hamming cubes $\{0,1\}^k = \{0,1\}^{k-1} \times \{0,1\}$. For all $n \geq 0$,
the interleaving distance between the prediction for $PH_n(X \times Y)$ and the
true persistent homology is bounded above by the minimum of the diameters of
$X$ and $Y$. As preliminary results of independent interest, we establish an
algebraic K\"unneth formula for simplicial modules over the ring
$\kappa[\mathbb{R}_+]$ of polynomials with coefficients in a field $\kappa$ and
exponents in $\mathbb{R}_+ = [0,\infty)$, as well as a K\"unneth formula for
the persistent homology of $\mathbb{R}_+$-filtered simplicial sets -- both of
these K\"unneth formulas hold in all homological dimensions $n \geq 0$.

探究度量空间的 Cartesian 积的 persistent homology 及其在 Kunneth 短正合序列、 interleaving distance 和 simplicial modules 方面的应用。

总度量的持久同调

Persistent homology of the sum metric

Unsupervised image-to-image translation aims at learning a mapping between
two visual domains. However, learning a translation across large geometry
variations always ends up with failure. In this work, we present a novel
disentangle-and-translate framework to tackle the complex objects
image-to-image translation task. Instead of learning the mapping on the image
space directly, we disentangle image space into a Cartesian product of the
appearance and the geometry latent spaces. Specifically, we first introduce a
geometry prior loss and a conditional VAE loss to encourage the network to
learn independent but complementary representations. The translation is then
built on appearance and geometry space separately. Extensive experiments
demonstrate the superior performance of our method to other state-of-the-art
approaches, especially in the challenging near-rigid and non-rigid objects
translation tasks. In addition, by taking different exemplars as the appearance
references, our method also supports multimodal translation. Project page:
this https URL

本文采用解缠和翻译框架解决复杂物体的图像翻译任务，使用外观和几何空间提取出独立但互补的表示，并分开翻译。实验结果表明，该方法在接近刚性和非刚性物体翻译任务中的表现优于其他最先进方法，同时支持多模式翻译。

TransGaGa：几何感知的无监督图像到图像翻译

TransGaGa: Geometry-Aware Unsupervised Image-to-Image Translation

We study the Ollivier-Ricci curvature of graphs as a function of the chosen
idleness. We show that this idleness function is concave and piecewise linear
with at most $3$ linear parts, with at most $2$ linear parts in the case of a
regular graph. We then apply our result to show that the idleness function of
the Cartesian product of two regular graphs is completely determined by the
idleness functions of the factors.

研究了作为所选空置时间函数的图的 Ollivier-Ricci 曲率，发现这个空置时间函数是凹的并且分段线性，最多有 $3$ 个线性部分，在正则图的情况下最多有 $2$ 个线性部分。然后应用我们的结果，证明了两个正则图笛卡尔积的空置时间函数完全由因子的空置时间函数确定。