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Andrea Schioppa

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math.MG math.DG math.CA Computer Vision Machine Learning

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preprint2026arXiv

Covariance-aware sampling for Diffusion Models

We present a covariance-aware sampler that improves the quality of pixel-space Diffusion Model (DM) sampling in the few-step regime. We hypothesize that in the few-step regime samplers fail because they rely solely on the predicted mean of the reverse distribution, while our solution explicitly models the reverse-process covariance. Our method combines Tweedie's formula to estimate the covariance with an efficient, structured Fourier-space decomposition of the covariance matrix. Implemented as an extension of DDIM, our method requires only a minimal overhead: one extra Jacobian-Vector Product (JVP) per step. We demonstrate that for pixel-based DMs, our method consistently produces superior samples compared to state-of-the-art second order samplers (Heun, DPM-Solver++) and the recent aDDIM sampler, at an identical number of function evaluations (NFE).

preprint2016arXiv

An example of a differentiability space which is PI-unrectifiable

We construct a (Lipschitz) differentiability space which has at generic points a disconnected tangent and thus does not contain positive measure subsets isometric to positive measure subsets of spaces admitting a Poincaré inequality. We also prove that $l^2$ -valued Lipschitz maps are differentiable a.e., but there are also Lipschitz maps taking values in some other Banach spaces having the Radon-Nikodym property which fail to be differentiable on sets of positive measure.

preprint2016arXiv

Derivations and Alberti representations

We relate generalized Lebesgue decompositions of measures in terms of curve fragments (Alberti representations) and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space $(X,μ)$ : the local norm of a form $df$ sees how fast $f$ grows on curve fragments seen by $μ$ . This implies a new characterization of differentiability spaces in terms of the $μ$ -a.e.~equality of the local norm of $df$ and the local Lipschitz constant of $f$ . As a consequence, the Lip-lip inequality of Keith must be an equality. We also provide dimensional bounds for the module of derivations in terms of the Assouad dimension of $X$ .

preprint2016arXiv

Examples of $2$ -unrectifiable normal currents

We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$ -dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$ -unrectifiable and have Nagata dimension $N$ . We show that in $l^\infty$ normal currents can be realized as limits in the flat distance of currents associated to cube complexes.

preprint2016arXiv

Infinitesimal structure of differentiability spaces, and metric differentiation

We prove metric differentiation for differentiability spaces in the sense of Cheeger. As corollaries we give a new proof that the minimal generalized upper gradient coincides with the pointwise Lipschitz constant for Lipschitz functions on PI spaces, a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith is equal to $1$ , and new nonembeddability results.

preprint2016arXiv

Metric Currents and Alberti representations

We relate Ambrosio-Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current $T$ , we show that if the module $\mathscr{X}(\|T\|)$ of Weaver derivations is finitely generated, then $T$ can be represented in terms of derivations; this extends previous results of Williams. Applications of this theory include an approximation of $1$ -dimensional metric currents in terms of normal currents and the construction of Alberti representations in the directions of vector fields.

preprint2016arXiv

The Poincaré Inequality does not improve with blow-up

For each $β>1$ we construct a family $F_β$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_β$ admits a $(1,P)$ -Poincaré inequality if and only if $P>β$ .

preprint2016arXiv

Unrectifiable normal currents in Euclidean spaces

We construct in $\mathbb{R}^{k+2}$ a $k$ -dimensional simple normal current whose support is purely $2$ -unrectifiable. The result is sharp because the support of a normal current cannot be purely $1$ -unrectifiable and a $(k+1)$ -dimensional normal current can be represented as an integral of $(k+1)$ -rectifiable currents. This gives a negative answer to the (revised version) of a question of Frank Morgan (1984).

preprint2015arXiv

PI spaces with analytic dimension 1 and arbitrary topological dimension

For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of Heinonen-Koskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.

preprint2015arXiv

The Lip-lip equality is stable under blow-up

We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$ -weak gradient on iterated blow-ups of differentiability spaces.

preprint2014arXiv

Poincaré inequalities for mutually singular measures

Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space $X$ that admits Poincaré inequalities for a continuum of mutually singular measures.

preprint2012arXiv

On the relationship between derivations and measurable differentiable structures on metric measure spaces

We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise upper Lipschitz constant of a function through derivations; [3] an extension of a result of Keith about the choice of chart functions.

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Andrea Schioppa

Research overview

Is this your work?

12 publications

Covariance-aware sampling for Diffusion Models

An example of a differentiability space which is PI-unrectifiable

Derivations and Alberti representations

Examples of $2$ -unrectifiable normal currents

Infinitesimal structure of differentiability spaces, and metric differentiation

Metric Currents and Alberti representations

The Poincaré Inequality does not improve with blow-up

Unrectifiable normal currents in Euclidean spaces

PI spaces with analytic dimension 1 and arbitrary topological dimension

The Lip-lip equality is stable under blow-up

Poincaré inequalities for mutually singular measures

On the relationship between derivations and measurable differentiable structures on metric measure spaces

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Andrea Schioppa

Research overview

Is this your work?

12 publications

Covariance-aware sampling for Diffusion Models

An example of a differentiability space which is PI-unrectifiable

Derivations and Alberti representations

Examples of 222-unrectifiable normal currents

Infinitesimal structure of differentiability spaces, and metric differentiation

Metric Currents and Alberti representations

The Poincaré Inequality does not improve with blow-up

Unrectifiable normal currents in Euclidean spaces

PI spaces with analytic dimension 1 and arbitrary topological dimension

The Lip-lip equality is stable under blow-up

Poincaré inequalities for mutually singular measures

On the relationship between derivations and measurable differentiable structures on metric measure spaces

See the researcher in context

Building this map preview

Examples of $2$ -unrectifiable normal currents