TL;DR: 1min.AI combines popular AI tools like GPT-4.0 and Midjourney, and lifetime access is only 79.97. AI tools like ChatGPT popped into existence, totally changed the professional world, and then immediately became very expensive. It's hard to get by without them now, but that doesn't mean you have to pay for each one individually. Instead of shelling out for OpenAI, Midjourney, and everything else, now you can get the same AI models all under one umbrella. This platform goes way beyond just text generation.

The original version of this story appeared in Quanta Magazine. Large language models work well because they're so large. The latest models from OpenAI, Meta, and DeepSeek use hundreds of billions of "parameters"--the adjustable knobs that determine connections among data and get tweaked during the training process. With more parameters, the models are better able to identify patterns and connections, which in turn makes them more powerful and accurate. But this power comes at a cost.

This paper addresses the problem of sparse phase retrieval, a fundamental inverse problem in applied mathematics, physics, and engineering, where a signal need to be reconstructed using only the magnitude of its transformation while phase information remains inaccessible. Leveraging the inherent sparsity of many real-world signals, we introduce a novel sparse quasi-Bayesian approach and provide the first theoretical guarantees for such an approach. Specifically, we employ a scaled Student distribution as a continuous shrinkage prior to enforce sparsity and analyze the method using the PAC-Bayesian inequality framework. Our results establish that the proposed Bayesian estimator achieves minimax-optimal convergence rates under sub-exponential noise, matching those of state-of-the-art frequentist methods. To ensure computational feasibility, we develop an efficient Langevin Monte Carlo sampling algorithm. Through numerical experiments, we demonstrate that our method performs comparably to existing frequentist techniques, highlighting its potential as a principled alternative for sparse phase retrieval in noisy settings.

Discovering constants of motion is meaningful in helping understand the dynamical systems, but inevitably needs proficient mathematical skills and keen analytical capabilities. With the prevalence of deep learning, methods employing neural networks, such as Constant Of Motion nETwork (COMET), are promising in handling this scientific problem. Although the COMET method can produce better predictions on dynamics by exploiting the discovered constants of motion, there is still plenty of room to sharpen it. In this paper, we propose a novel neural network architecture, built using the singular-value-decomposition (SVD) technique, and a two-phase training algorithm to improve the performance of COMET. Extensive experiments show that our approach not only retains the advantages of COMET, such as applying to non-Hamiltonian systems and indicating the number of constants of motion, but also can be more lightweight and noise-robust than COMET.

We introduce aweSOM, an open-source Python package for machine learning (ML) clustering and classification, using a Self-organizing Maps (SOM) algorithm that incorporates CPU/GPU acceleration to accommodate large ($N > 10^6$, where $N$ is the number of data points), multidimensional datasets. aweSOM consists of two main modules, one that handles the initialization and training of the SOM, and another that stacks the results of multiple SOM realizations to obtain more statistically robust clusters. Existing Python-based SOM implementations (e.g., POPSOM, Yuan (2018); MiniSom, Vettigli (2018); sklearn-som) primarily serve as proof-of-concept demonstrations, optimized for smaller datasets, but lacking scalability for large, multidimensional data. aweSOM provides a solution for this gap in capability, with good performance scaling up to $\sim 10^8$ individual points, and capable of utilizing multiple features per point. We compare the code performance against the legacy implementations it is based on, and find a 10-100x speed up, as well as significantly improved memory efficiency, due to several built-in optimizations.

In this paper, we study the offline sequential feature-based pricing and inventory control problem where the current demand depends on the past demand levels and any demand exceeding the available inventory is lost. Our goal is to leverage the offline dataset, consisting of past prices, ordering quantities, inventory levels, covariates, and censored sales levels, to estimate the optimal pricing and inventory control policy that maximizes long-term profit. While the underlying dynamic without censoring can be modeled by Markov decision process (MDP), the primary obstacle arises from the observed process where demand censoring is present, resulting in missing profit information, the failure of the Markov property, and a non-stationary optimal policy. To overcome these challenges, we first approximate the optimal policy by solving a high-order MDP characterized by the number of consecutive censoring instances, which ultimately boils down to solving a specialized Bellman equation tailored for this problem. Inspired by offline reinforcement learning and survival analysis, we propose two novel data-driven algorithms to solving these Bellman equations and, thus, estimate the optimal policy. Furthermore, we establish finite sample regret bounds to validate the effectiveness of these algorithms. Finally, we conduct numerical experiments to demonstrate the efficacy of our algorithms in estimating the optimal policy. To the best of our knowledge, this is the first data-driven approach to learning optimal pricing and inventory control policies in a sequential decision-making environment characterized by censored and dependent demand. The implementations of the proposed algorithms are available at https://github.com/gundemkorel/Inventory_Pricing_Control

I show that ordinary least squares (OLS) predictions can be rewritten as the output of a restricted attention module, akin to those forming the backbone of large language models. This connection offers an alternative perspective on attention beyond the conventional information retrieval framework, making it more accessible to researchers and analysts with a background in traditional statistics. It falls into place when OLS is framed as a similarity-based method in a transformed regressor space, distinct from the standard view based on partial correlations. In fact, the OLS solution can be recast as the outcome of an alternative problem: minimizing squared prediction errors by optimizing the embedding space in which training and test vectors are compared via inner products. Rather than estimating coefficients directly, we equivalently learn optimal encoding and decoding operations for predictors. From this vantage point, OLS maps naturally onto the query-key-value structure of attention mechanisms. Building on this foundation, I discuss key elements of Transformer-style attention and draw connections to classic ideas from time series econometrics.

Testing conditional independence between two random vectors given a third is a fundamental and challenging problem in statistics, particularly in multivariate nonparametric settings due to the complexity of conditional structures. We propose a novel framework for testing conditional independence using transport maps. At the population level, we show that two well-defined transport maps can transform the conditional independence test into an unconditional independence test, this substantially simplifies the problem. These transport maps are estimated from data using conditional continuous normalizing flow models. Within this framework, we derive a test statistic and prove its consistency under both the null and alternative hypotheses. A permutation-based procedure is employed to evaluate the significance of the test. We validate the proposed method through extensive simulations and real-data analysis. Our numerical studies demonstrate the practical effectiveness of the proposed method for conditional independence testing.

Layer-wise post-training quantization has emerged as a widely used technique for compressing large language models (LLMs) without retraining. However, recent progress in this line of research is saturating, underscoring the need to revisit its core limitation and explore further improvements. This study identifies a critical bottleneck in existing layer-wise PTQ methods: the accumulation of quantization errors across layers significantly degrades performance, particularly in low-bit regimes. To address this, we propose Quantization Error Propagation (QEP), a lightweight and general framework that enhances layer-wise PTQ by explicitly propagating the quantization error which enable compensating for accumulated quantization errors. Additionally, we introduce a tunable propagation mechanism that allows for control over both propagation strength and computational overhead, making the framework adaptable to various architectures and resource constraints. Empirical evaluation on LLaMA2 models (7B, 13B, 70B) demonstrate that incorporating QEP into standard layer-wise PTQ pipelines outperforms standard PTQ methods. Notably, QEP yields substantial performance improvements under extreme low-bit quantization settings.

In practical instances of nonconvex matrix factorization, the rank of the true solution $r^{\star}$ is often unknown, so the rank $r$ of the model can be overspecified as $r>r^{\star}$. This over-parameterized regime of matrix factorization significantly slows down the convergence of local search algorithms, from a linear rate with $r=r^{\star}$ to a sublinear rate when $r>r^{\star}$. We propose an inexpensive preconditioner for the matrix sensing variant of nonconvex matrix factorization that restores the convergence rate of gradient descent back to linear, even in the over-parameterized case, while also making it agnostic to possible ill-conditioning in the ground truth. Classical gradient descent in a neighborhood of the solution slows down due to the need for the model matrix factor to become singular. Our key result is that this singularity can be corrected by $\ell_{2}$ regularization with a specific range of values for the damping parameter. In fact, a good damping parameter can be inexpensively estimated from the current iterate. The resulting algorithm, which we call preconditioned gradient descent or PrecGD, is stable under noise, and converges linearly to an information theoretically optimal error bound. Our numerical experiments find that PrecGD works equally well in restoring the linear convergence of other variants of nonconvex matrix factorization in the over-parameterized regime.