Hadamard's lemma in separable Hilbert spaces
We extend Hadamard's Lemma to the setting of a separable Hilbert space.
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arXiv subjects
Arian Bërdëllima
Publications and source records attributed to Arian Bërdëllima.
Banach spaces of sequences arising from infinite matrices
Given an infinite matrix $M=(m_{nk})$ we study a family of sequence spaces $\ell_M^p$ associated with it. When equipped with a suitable norm $\|\cdot\|_{M,p}$ we prove some basic properties of the Banach spaces of sequences $(\ell_M^p,\|\cdot\|_{M,p})$. In particular we show that such spaces are separable and strictly/uniformly convex for a considerably large class of infinite matrices $M$ for all $p>1$. A special attention is given to the identification of the dual space $(\ell_M^p )^*$. Building on the earlier works of Bennett and Jägers, we extend and apply some classical factorization results to the sequence spaces $\ell_M^p$.
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Existence and uniqueness of optimal transport maps in locally compact $CAT(0)$ spaces
We show that in a locally compact complete $CAT(0)$ space satisfying positive angles property and a disintegration regularity for its canonical Hausdorff measure, there exists a unique optimal transport map that push-forwards a given absolutely continuous probability measure to another probability measure. In particular this holds for the Riemannian manifolds of non-positive sectional curvature and $CAT(0)$ Euclidean polyhedral complexes. Moveover we give a polar factorization result for Borel maps in $CAT(0)$ spaces in terms of optimal transport maps and measure preserving maps.
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On Nörlund summability of Taylor series in weighted Dirichlet spaces
In this note we show that the Taylor series of a function in a weighted Dirichlet space is (generalized) Nörlund summable, provided that the sequence determining the Nörlund operator is non-decreasing and has finite upper growth rate. In particular the Taylor series is Nörlund summable for all $α>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The inequality $α>1/2$ is sharp. On the other hand if the Taylor series is Nörlund summable and the partial sums of the determining sequence enjoy a certain growth condition then the determining sequence has finite lower growth rate. An analogue result is derived for a non-increasing sequence that is uniformly bounded away from zero.
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Quasi $α$-Firmly Nonexpansive Mappings in Wasserstein Spaces
This paper introduces the concept of quasi $α$-firmly nonexpansive mappings in Wasserstein spaces over $\mathbb R^d$ and analyzes properties of these mappings. We prove that for quasi $α$-firmly nonexpansive mappings satisfying a certain quadratic growth condition, the fixed point iterations converge in the narrow topology. As a byproduct, we will get the known convergence of the proximal point algorithm in Wasserstein spaces. We apply our results to show for the first time that cyclic proximal point algorithms for minimizing the sum of certain functionals on Wasserstein spaces converge under appropriate assumptions.
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On a weak topology for Hadamard spaces and its applications
We investigate if an existing notion of weak sequential convergence in a Hadamard space can be induced by a topology. We provide an answer on what we call weakly proper Hadamard spaces. A notion of dual space is proposed and it is shown that our weak topology and dual space coincide with the standard ones in the case of a Hilbert space. Moreover we introduce the space of geodesic segments and a corresponding weak topology, and we show that this space is homeomorphic to its underlying Hadamard space. As an application of it we show the existence of a geodesic segment that acts as direction of steepest descent for a geodesically differentiable function whose geodesic derivative satisfies certain properties. Finally we extend several results from classical functional analysis to the setting of Hadamard spaces, and we compare our topology with other existing notions of weak topologies.
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Compact sets and the closure of their convex hulls in CAT(0) spaces
We study the closure of the convex hull of a compact set in a complete CAT(0) space. First we give characterization results in terms of compact sets and the closure of their convex hulls for locally compact CAT(0) spaces that are either regular or satisfy the geodesic extension property. Later inspired by a geometric interpretation of Carathéodory's Theorem we introduce the operation of threading for a given set. We show that threading exhibits certain monotonicity properties with respect to intersection and union of sets. Moreover threading preserves compactness. Next from the commutativity of threading with any isometry mapping we prove that in a flat complete CAT(0) space the closure of the convex hull of a compact set is compact. We apply our theory to the computability of the Fréchet mean of a finite set of points and show that it is constructible in at most a finite number of steps, whenever the underlying space is of finite type.
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On the Dynamical System of Principal Curves in $\mathbb R^d$
Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based on the conditional expectation and from the variational-calculus point of view as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. Beyond that, Duchamp and Stützle (1993,1996) showed that planar curves can by computed as solutions of a system of ordinary differential equations. The aim of this paper is to generalize this characterization of principal curves to $\mathbb R^d$ with $d \ge 3$. Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in $\mathbb R^3$.
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On sharp constants in Paley problem for plurisubharmonic functions of lower order $ρ>1$
In 1999 Khabibullin established the best estimate in Paley problem for a plurisubharmonic function $u$ of finite lower order $0\leqρ\leq 1$. For $ρ>1$ obtaining a sharp estimate has remained an open question. In this work we solve this problem. We also provide some estimates for the types of the characteristic functions $T(r,u)$ and $M(r,u)$.
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$α$-Firmly Nonexpansive Operators on Metric Spaces
We extend to $p$-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise $α$-averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying $α$-firm nonexpansiveness of the individual components at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that only $α$-firm nonexpansiveness with respect to fixed points is required. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.
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On $α$-Firmly Nonexpansive Operators in $r$-Uniformly Convex Spaces
We introduce the class of $α$-firmly nonexpansive and quasi $α$-firmly nonexpansive operators on $r$-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where $α$-firmly nonexpansive operators coincide with so-called $α$-averaged operators. For our more general setting, we show that $α$-averaged operators form a subset of $α$-firmly nonexpansive operators. We develop some basic calculus rules for (quasi) $α$-firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) $α$-firmly nonexpansive. Moreover, we will see that quasi $α$-firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder's demiclosedness principle, we prove for $r$-uniformly convex Banach spaces that the weak cluster points of the iterates $x_{n+1}:=Tx_{n}$ belong to the fixed point set $\text{Fix} T$ whenever the operator $T$ is nonexpansive and quasi $α$-firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial's property then these iterates converge weakly to some element in $\text{Fix} T$. Further, the projections $P_{\text{Fix} T}x_n$ converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in $L_p$, $p \in (1,\infty) \backslash \{2\}$ spaces on probability measure spaces.
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