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Warning. The reading of this paper will send you down many winding roads toward new and exciting research topics enumerating generalized parking functions. Buckle up!
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We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of (m,n)-parking functions, and associated (q,t)-analogs.
Can a user create a deep generative model by sketching a single example? Traditionally, creating a GAN model has required the collection of a large-scale dataset of exemplars and specialized knowledge in deep learning. In contrast, sketching is possibly the most universally accessible way to convey a visual concept. In this work, we present a method, GAN Sketching, for rewriting GANs with one or more sketches, to make GANs training easier for novice users. In particular, we change the weights of an original GAN model according to user sketches. We encourage the models output to match the user sketches through a cross-domain adversarial loss. Furthermore, we explore different regularization methods to preserve the original models diversity and image quality. Experiments have shown that our method can mold GANs to match shapes and poses specified by sketches while maintaining realism and diversity. Finally, we demonstrate a few applications of the resulting GAN, including latent space interpolation and image editing.
Geometric feature extraction is a crucial component of point cloud registration pipelines. Recent work has demonstrated how supervised learning can be leveraged to learn better and more compact 3D features. However, those approaches reliance on ground-truth annotation limits their scalability. We propose BYOC: a self-supervised approach that learns visual and geometric features from RGB-D video without relying on ground-truth pose or correspondence. Our key observation is that randomly-initialized CNNs readily provide us with good correspondences; allowing us to bootstrap the learning of both visual and geometric features. Our approach combines classic ideas from point cloud registration with more recent representation learning approaches. We evaluate our approach on indoor scene datasets and find that our method outperforms traditional and learned descriptors, while being competitive with current state-of-the-art supervised approaches.
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haimans notion of a parking function symmetric function.
The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the nth Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.
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