We expect it to leak so we're just announcing it. Exactly a week after Anthropic announced its plan to go public, OpenAI has followed suit. The company said on Monday that it confidentially submitted a S-1 form with the Securities and Exchange Commission. No date or offer price has been set by OpenAI yet for the initial public offering. We recently submitted a confidential S-1. We expect it to leak so we're just announcing it.

The subscription now starts at $5 per month. Google is lowering the cost of its cheapest AI subscription to make Gemini models even easier to access. The Google AI Plus plan will now cost $5 per month, according to a post from Vikas Kansal, the company's Product Lead focused on Gemini AI subscriptions, down from its original $8 per month price. It now also comes with double the storage, 400GB instead of 200GB. The subscription plan became available in January 2026 as a cheaper way to access Google's Gemini 3 Pro model, Nano Banana Pro and Deep Research.

The Download: how the World Cup ball will fly and OpenAI's "super app" Plus: OpenAI plans to turn ChatGPT into a'super app' before its IPO. Why this year's World Cup ball may not fly as far Much is new about this month's FIFA World Cup tournament. It hosts more teams than ever before. It's the first to occur in three different host countries. And, like every World Cup for over half a century, it will employ a football with a brand-new design. Through wind-tunnel experiments, researchers found that long-distance kicks with Adidas's new Trionda ball might not travel as far as they did in the past.

A central mystery in deep learning is how neural networks, despite being highly non-convex and heavily overparameterized, are able to achieve near-zero training error while still generalizing well to unseen data. This paradox has sparked a surge of research aimed at understanding the convergence and generalization behavior of neural networks [1, 2, 6, 7, 15, 38, 41, 49]. The Neural Tangent Kernel (NTK), introduced by [20], has become one of a foundational tool for understanding the behavior of training dynamics for neural networks, especially those trained using gradient-based methods such as gradient descent (GD) and stochastic gradient descent (SGD). The core idea here is to linearize the neural network around its random initialization, which enables the evolution of the network during training to be closely approximated by a kernel method associated with the corresponding NTK. This framework establishes a powerful connection between the evolution of a neural network during training process and the behavior of kernel methods in a reproducing kernel Hilbert space (RKHS) induced by the NTK, allowing insights from the kernel methods to inform our understanding of neural networks. Following this perspective, the influential work [34] showed that for regression problems, shallow neural networks trained by SGD can achieve generalization performance on par with their kernel counterparts.

Predicting outputs that are located in non-Euclidean spaces, such as probability distributions, networks, and symmetric positive-definite matrices, is becoming increasingly important in modern data analysis, particularly when inputs are high-dimensional. We propose DeSI (Deep Single-Index Fréchet Regression), a semiparametric framework for regression with metric space-valued outputs and multivariate inputs that assumes a single-index structure for the conditional Fréchet mean. DeSI estimates an interpretable index direction, which quantifies the relative importance of inputs, using a deep neural network, and performs Fréchet regression along the resulting one-dimensional index in the target metric space. This structure mitigates the curse of dimensionality while retaining interpretability, which stands in contrast to standard deep neural networks. We establish theoretical guarantees for DeSI, including uniform approximation and convergence rates, and demonstrate its strong predictive performance through simulations on distributions, networks, and symmetric positive-definite matrices, as well as an application to compositional mood data from New Jersey.

Recent progress has been made in understanding the statistical generalization performance of gradient descent methods for overparameterized neural networks within the neural tangent kernel (NTK) regime. However, most of the existing work on regression problems is limited to shallow network architectures, leaving a notable gap in the theory of deep neural networks. This paper addresses this gap by presenting a comprehensive generalization analysis for deep ReLU networks trained using gradient descent (GD) and stochastic gradient descent (SGD). Specifically, we establish the first known minimax-optimal rates of excess population risk for both GD and SGD with deep ReLU networks, under the assumption that the network width scales polynomially with respect to the network depth and training sample size. Our results demonstrate that with sufficient width, gradient descent methods for deep ReLU networks can achieve optimal generalization rates on par with kernel methods.

He is about to take SpaceX public--pushing other AI companies to do the same. Elon Musk is about to set in motion a chain of events that will reshape the global financial order. For starters, when SpaceX formally goes public next week, he is all but guaranteed to become the world's first trillionaire. His rocket company is targeting a valuation of $1.77 trillion, which would make it one of the 10 biggest companies in the world--bigger than Meta, Walmart, and, for that matter, Tesla. All of this activity is less about colonizing Mars and more about providing the infrastructure for the AI boom: Musk wants to use his rockets to launch data centers into space, where there is abundant solar power to harvest.

Plus: Anthropic has called for a global slowdown in AI development. The Meta hack shows there's more to AI security than Mythos On Monday, reports emerged that attackers had used Meta's AI customer support agent to steal Instagram accounts. Their approach was simple: they asked the agent to link the accounts to email addresses they controlled, and it complied. Since Anthropic announced that its Mythos model was too good at hacking for a general release, cybersecurity concerns have focused on the risk of superpowered AI systems overwhelming computer infrastructure. But the Instagram hack shows that far simpler exploits can still cause damage. As companies offload more work to AI, these comparatively unsophisticated attacks are becoming harder to ignore.

Singular learning theory and information geometry have studied the same parameter spaces in mostly separate vocabularies: the former computes Bayesian invariants in resolved coordinates, the latter works in original coordinates under a non-degeneracy assumption that overparameterised models routinely violate. We bridge them through one primitive, the dead direction: a unit vector along which the Fisher metric degenerates, equivalently a tangent to the analytic singular set with a definite KL order, set by how fast the KL divergence vanishes. The two readings name the same vector; our central move shows its KL order is recoverable as the decay rate of the directional Fisher curvature approaching the singularity, in original parameter coordinates and without a Hironaka resolution. A selection rule on smooth fibres translates this rate into Watanabe's single-direction contribution to the real log canonical threshold, and we extend the recovery to multi-component crossings, multiplicity $m$, the singular fluctuation $ν$ (universal in the KL order for 1D directions), prior-RLCT shifts, and tempered posteriors. We then lift this rate to a deep network: a multi-layer K-FAC factorisation writes each Fisher block as a product of activation- and gradient-side rates with a duality between them, instantiated at modern-network primitives (residual streams, layer normalisation, attention). A quotient theorem carries the rate to the gauge quotient $Θ/G$ under gradient flow on a $G$-invariant metric; SGD qualifies, standard Adam does not, and we construct a $G$-equivariant Adam-family preconditioner (DDCAdam) that does. The bridge yields a parameter-coordinate handle on singular geometry, closed-form per-architecture predictions, and a trajectory-rate readout of Watanabe's triple $(λ, m, ν)$ from one checkpoint's forward and backward passes, without posterior sampling.

Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of geometry-aware machine learning algorithms, such as the Mean Curvature Boundary Points (MCBP) method. The naive implementation of this computation, based on a local shape operator approximated from k-nearest neighbor patches, involves an explicit construction of a matrix $H$ whose trace form yields an $O(m^4)$ cost per point, rendering the approach intractable for datasets with more than a few dozen features. This paper introduces two complementary contributions that together reduce this cost by several orders of magnitude. The first contribution is an exact algebraic identity. This identity, derived from the orthogonality of the eigenvectors of the covariance matrix and the cyclicity of the trace operator, eliminates $H$ entirely and reduces the per-point cost to $O(m^2)$ after the eigendecomposition. The second contribution addresses the remaining $O(m^3)$ bottleneck of the full eigendecomposition. Since the local covariance matrix has rank at most $k-1 \ll m$, we replace it with a truncated SVD of the $k \times m$ centered data matrix, an $O(k^2 m)$ operation, and derive an analytical approximation for the contribution of the null-space eigenvectors based on the expected value of their outer product under the Haar measure. The resulting estimator has total cost $O(k^2 m + k m p^2)$, where $p = k-1$. Experiments on real-world datasets confirm speedups of 50 to 300 times relative to the original implementation, with negligible loss when the fast estimator is used to replace the original version. By providing a scalable and data-driven estimate of local curvature, the proposed method establishes curvature as a practical geometric feature for a broad range of machine learning tasks, from classical to modern deep learning pipelines.