The concept of p-th variation of a real-valued continuous function along a general class of refining sequence of partitions is presented. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to ℓ∞-norm of the Schauder coefficients. This result provides an isomorphism between the space of α-Hölder continuous functions with finite (generalized) p-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.

We propose a general learning based framework for solving nonsmooth and nonconvex inverse problems with application to low-dose CT (LDCT) reconstruction. We model the regularization function as the combination of a sparsity enhancing and a non-local smoothing regularization. We develop an efficient learned descent-type algorithm (ELDA) to solve the nonsmooth nonconvex minimization problem by leveraging the Nesterov’s smoothing technique and incorporating the residual learning structure. We proved the convergence of the algorithm and generate the network, whose architecture follows the algorithm exactly. Our method is versatile as one can employ various modern network structures into the regularization, and the resulting network inherits the convergence properties, and hence is interpretable. We also show that the proposed network is parameter-efficient and its performance compares favorably to the state-of-the-art methods.

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https://kaist.zoom.us/j/82680768716?pwd=4jDj5hW70PKYbTcYq1nbkEa9Gsarhi.1 Meeting ID: 826 8076 8716 Passcode: 933841 참고: Jan 16, 2025 07:00 PM Eastern Time (US and Canada) https://kaist.zoom.us/j/82680768716?pwd=4jDj5hW70PKYbTcYq1nbkEa9Gsarhi.1 Meeting ID: 826 8076 8716 Passcode: 933841

Given a manifold, the vertices of a geometric intersection graph are defined as a class of submanifolds. Whether there is an edge between two vertices depends on their geometric intersection numbers. The geometric intersection complex is the clique complex induced by the geometric intersection graph. Common examples include the curve (arc) complex and the Kakimizu complex. Curve complexes and arc complexes are used to understand mapping class groups and Teichmüller spaces, while Kakimizu complexes are primarily used to study hyperbolic knots. We can study these geometric intersection complexes from various perspectives, including topology (e.g., homotopy type), geometry (e.g., dimension, diameter, hyperbolicity), and number-theoretic connections (e.g., trace formulas of maximal systems). In this talk, we will mainly explain how to determine the dimension of the (complete) $1$-curve (or arc) complex on a non-orientable surface and examine the transitivity of maximal complete $1$-systems of loops on a punctured projective plane.