Search

Conventional nonlinear RNNs are not naturally parallelizable across the sequence length, unlike transformers and linear RNNs. Lim et. al. (2024) therefore tackle parallelized evaluation of nonlinear RNNs, posing it as a fixed point problem solved with Newton's method. By deriving and applying a parallelized form of Newton's method, they achieve large speedups over sequential evaluation. However, their approach inherits cubic computational complexity and numerical instability. We tackle these weaknesses. To reduce the computational complexity, we apply quasi-Newton approximations and show they converge comparably, use less memory, and are faster, compared to full-Newton. To stabilize Newton's method, we leverage a connection between Newton's method damped with trust regions and Kalman smoothing. This connection allows us to stabilize the iteration, per the trust region, and use efficient parallelized Kalman algorithms to retain performance. We compare these methods empirically and highlight use cases where each algorithm excels., Comment: 25 pages, 8 figures, NeurIPS 2024

Lines are human mental abstractions. A bunch of lines may form a drawing. A set of drawings can feed an LSTM network input layer, considering each draw as a list of lines and a line a list of points. This paper proposes the pointless motive to classify the gender of celebrities' portraits as an excuse for exploration in a broad, more artistic sense. Investigation results drove compelling ideas here discussed. The experiments compared different ways to represent draws to be input in a network and showed that an absolute format of coordinates (x, y) was a better performer than a relative one (Dx, Dy) with respect to prior points, most frequent in the reviewed literature. Experiments also showed that, due to the recurrent nature of LSTMs, the order of lines forming a drawing is a relevant factor for input in an LSTM classifier not studied before. A minimum 'pencil' traveled length criteria for line ordering proved suitable, possible by reducing it to a TSP particular instance. The best configuration for gender classification appears with an LSTM layer that returns the hidden state value for each input point step, followed by a global average layer along the sequence, before the output dense layer. That result guided the idea of removing the average in the network pipeline and return a per-point attribute score just by adjusting tensors dimensions. With this trick, the model detects an attribute in a drawing and also recognizes the points linked to it. Moreover, by overlapping filtered lines of portraits, an attribute's visual essence is depicted. Meet the FaCells.

A crypto coin designed to provide a stabilization instrument backed up by minded like financial investments instruments to maintain the purchase value of savings across time, in order to construct new tools for unstable economies., Comment: 14 pages, 3 figures

We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, marking the birth of the "circle method", we present a contemporary example of its legacy in topology. We deduce the equidistribution of Hodge numbers for Hilbert schemes of suitable smooth projective surfaces., Comment: 14 pages. Minor edits. To appear in the a Philosophical Transactions of the Royal Society

A theorem of G\"ottsche establishes a connection between cohomological invariants of a complex projective surface $S$ and corresponding invariants of the Hilbert scheme of $n$ points on $S.$ This relationship is encoded in certain infinite product $q$-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these $q$-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the $q$-series' coefficients., Comment: 25 pages, 1 figure, Accepted for publication in Research in Number Theory

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group $G$, we give constructions of infinitely many graded infinite-dimensional $\mathbb{C}[G]$-modules where the McKay-Thompson series for a conjugacy class $[g]$ is a weakly holomorphic modular function properly on $\Gamma_0(\text{ord}(g))$. As there are only finitely many normalized Hauptmoduln, groups whose McKay-Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level $\text{ord}(g)$ as the graded trace functions for any conjugacy class $[g]$, and completely classify the finite abelian groups with this property. In particular, these include $(\mathbb{Z} / 5 \mathbb{Z})^5$ and $(\mathbb{Z} / 7 \mathbb{Z})^4$, which are not subgroups of the Monster., Comment: 34 pages