A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We prove a scaling limit theorem for the super-replication cost of options in a Cox--Ross--Rubinstein binomial model with transient price impact. The correct scaling turns out to keep the market depth parameter constant while resilience over fixed periods of time grows in inverse proportion with the duration between tr…
Training very deep networks is an important open problem in machine learning. One of many difficulties is that the norm of the back-propagated error gradient can grow or decay exponentially. Here we show that training very deep feed-forward networks (FFNs) is not as difficult as previously thought. Unlike when back-pro…
This paper explores how neural network width and depth behave as they approach infinity.
problem Understanding the behavior of neural functions as width and depth go to infinity.
method Formal definition of commutativity framework, study of neural covariance kernel, novel proof techniques.
result Taking width and depth to infinity in a deep neural network with skip connections results in the same covariance structure, regardless of the order of taking limits.
Unified learning-rate scale for CNNs and ResNets, avoiding depth imbalance.
problem Challenges in choosing an appropriate learning rate for deep networks, especially as depth increases.
method Introduces Arithmetic-Mean μP (AM-μP), constraining network-wide average pre-activation second moment to a constant scale, combined with residual-aware He fan-in initialization.
result Demonstrates a −3/2 scaling law for learning rates across depths, enabling zero-shot learning-rate transfer.
Deep networks with orthogonal weights show stable fluctuations, improving generalization and training speed.
problem Fluctuations in deep networks with Gaussian weights can impair training, especially in networks with depth comparable to width.
method Analytical and numerical studies of fully-connected networks with orthogonal weight initialization and tanh activations.
result Rectangular networks with orthogonal weights have stable fluctuations independent of network depth, leading to better generalization and training speed.
We study the behavior of untrained neural networks whose weights and biases are randomly distributed using mean field theory. We show the existence of depth scales that naturally limit the maximum depth of signal propagation through these random networks. Our main practical result is to show that random networks may be…
We study the price impact of order book events - limit orders, market orders and cancelations - using the NYSE TAQ data for 50 U.S. stocks. We show that, over short time intervals, price changes are mainly driven by the order flow imbalance, defined as the imbalance between supply and demand at the best bid and ask pri…
Existing depth separation results for constant-depth networks essentially show that certain radial functions in Rd, which can be easily approximated with depth 3 networks, cannot be approximated by depth 2 networks, even up to constant accuracy, unless their size is exponential in d. However, the func…
In this paper we perform a statistical analysis of the high-frequency returns of the IBEX35 Madrid stock exchange index. We find that its probability distribution seems to be stable over different time scales, a stylized fact observed in many different financial time series. However, an in-depth analysis of the data us…
We show that the number of unique function mappings in a neural network hypothesis space is inversely proportional to ∏lUl!, where Ul is the number of neurons in the hidden layer l.
Recent work in signal propagation theory has shown that dropout limits the depth to which information can propagate through a neural network. In this paper, we investigate the effect of initialisation on training speed and generalisation for ReLU networks within this depth limit. We ask the following research question:…
Inverse problems in imaging such as denoising, deblurring, superresolution (SR) have been addressed for many decades. In recent years, convolutional neural networks (CNNs) have been widely used for many inverse problem areas. Although their indisputable success, CNNs are not mathematically validated as to how and what …
We study inverse problems consisting on determining medium properties using the responses to probing waves from the machine learning point of view. Based on the understanding of propagation of waves and their nonlinear interactions, we construct a deep convolutional neural network in which the parameters are used to cl…
Proposes an online method for high-dimensional streaming data.
problem Increasing variable dimensions with sample size in online kernel sliced inverse regression.
method Introduces approximate linear dependence condition and dictionary variable sets to address the problem. Transforms into online generalized eigen-decomposition problem and uses stochastic optimization for updates.
result Achieves close performance to batch processing kernel sliced inverse regression.
Iterative learning to infer approaches have become popular solvers for inverse problems. However, their memory requirements during training grow linearly with model depth, limiting in practice model expressiveness. In this work, we propose an iterative inverse model with constant memory that relies on invertible networ…
New scaling framework for MoE architectures ensures stability and optimal performance at scale.
problem Lack of principled understanding of how hyperparameters should scale in MoE architectures.
method Developed a novel Dynamical Mean Field Theory (DMFT) for three scaling regimes of MoE architectures.
result Derived Maximally Scale-Stable Parameterization (MSSP) for SGD and Adam, providing robust learning rate transfer and monotonic improvement with scale.
Study reveals how Fisher information changes with network depth, finding it grows linearly.
problem Understanding the trainability of deep neural networks (DNNs).
method Investigates the spectral distribution of the conditional Fisher information matrix (FIM) for fully-connected networks achieving dynamical isometry.
result The conditional FIM's spectrum concentrates around the maximum and grows linearly with depth.
Per-pixel ground-truth depth data is challenging to acquire at scale. To overcome this limitation, self-supervised learning has emerged as a promising alternative for training models to perform monocular depth estimation. In this paper, we propose a set of improvements, which together result in both quantitatively and …
We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not determinist…
It is well known that the problem of vanishing/exploding gradients is a challenge when training deep networks. In this paper, we describe another phenomenon, called vanishing nodes, that also increases the difficulty of training deep neural networks. As the depth of a neural network increases, the network's hidden node…
Proves rigidity of circle packings in the plane, generalizing previous work.
problem Rigidity of infinite inversive distance circle packings in the plane.
method Maximal principle for generic weighted Delaunay inversive distance circle packings and ring lemma for inversive distance circle packings in hexagonal triangulated plane.
result Proves Bowers-Stephenson's conjecture for inversive distance circle packings.