In this paper, we introduce the notion of DTM-signature, a measure on R+ that can be associated to any metric-measure space. This signature is based on the distance to a measure (DTM) introduced by Chazal, Cohen-Steiner and Mérigot. It leads to a pseudo-metric between metric-measure spaces, upper-bounded by the Gromov-Wasserstein distance. Under some geometric assumptions, we derive lower bound...

Let l be a length function on a group G, and let Ml denote the operator of pointwise multiplication by l on l(G). Following Connes, Ml can be used as a “Dirac” operator for C ∗ r (G). It defines a Lipschitz seminorm on C∗ r (G), which defines a metric on the state space of C∗ r (G). We investigate whether the topology from this metric coincides with the weak-∗ topology (our definition of a “com...

A new method for solving domain equations in categories of metric spaces is studied. The categories CMS≈ and KMS≈ are introduced, having complete and compact metric spaces as objects and -adjoint pairs as arrows. The existence and uniqueness of fixed points for certain endofunctors on these categories is established. The classes of complete and compact metric spaces are considered as pseudo-met...

We study a generalization of the Bakry-\'Emery pointwise gradient estimate for heat semigroup and its equivalence with some entropic inequalities along flow Wasserstein geodesics metric-measure spaces suitable group structure. Our main result applies to Carnot groups any step $\mathbb{SU}(2)$ group.

Recently, some mixture algorithms of pointwise and pairwise learning (PPL) have been formulated by employing the hybrid error metric “pointwise loss + loss” shown empirical effectiveness on feature selection, ranking recommendation tasks. However, to best our knowledge, theory foundation PPL has not touched in existing works. In this paper, we try fill theoretical gap investigating generalizati...

A new method for solving domain equations in categories of metric spaces is studied. The categories CMS and KMS are introduced, having complete and compact metric spaces as objects and-adjoint pairs as arrows. The existence and uniqueness of xed points for certain endofunctors on these categories is established. The classes of complete and compact metric spaces are considered as pseudo-metric s...