'Ringleader' of alleged UFC drone attack to kill Trump is unmasked as illegal migrant who was granted DACA stay under Obama Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Horrific new videos blow Texas woman's mystery death wide open: Her agonizing'final gasp'... unthinkably vile corpse claims... and sick past of man who saw her last Taylor Swift's bottomless thirst for attention, her greed and sheer tackiness are now truly unbearable... this latest stunt has shown her true colors: MAUREEN CALLAHAN Spy world panic as Tulsi Gabbard prepares to unleash bombshell file dumps on secret CIA'mind control' project and Dr. Fauci Olivia Wilde, 42, complains about being on Maxim's Hot 100 List calling it the'most f***** up thing in the world' Has Taylor Swift already revealed her wedding dress designer? All my friends are suddenly getting divorced. Mid-life wives share taboo sex confessions about why they really leave... including common position that made one hate her husband: JANA HOCKING Kanye West's wife Bianca Censori raises eyebrows in plunging white lace lingerie as she photographs a nude model at Art Basel in Switzerland Knicks set to come face to face with Trump after president was'thunderously booed' at NBA Finals game Sensational REAL reason Jelly Roll is divorcing Bunnie XO: Insiders reveal'preacher's wife' bombshell that's the talk of Nashville... truth about legendary rocker cuckolding rumor... and G-string mishap Teen tourist thrown to death by Central Park horse was trying to save mom who flew out of carriage during family's first visit to Big Apple Father keeps his cool as shouting man calls cops on him for taking his two young daughters into women's restroom Trump privately frets Bibi Netanyahu's zeal to'bomb everyone' could turn him into another disgraced president'Moscow will burn', Zelensky vows as Russia's capital is blanketed in toxic smoke following huge Ukraine drone attack He drove a Rolls-Royce and lived the American dream. But behind the Gucci was the ATF's most unlikely secret weapon. MORE: Meteorologist reveals America's most dangerous cities in super El Niño's'corridor of chaos'... and warns this is only the beginning As many as 40 million people across eight states are in the deadly path of Tropical Storm Arthur after the first named storm of hurricane season made landfall Wednesday night.

'Ringleader' of alleged UFC drone attack to kill Trump is unmasked as illegal migrant who was granted DACA stay under Obama Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Horrific new videos blow Texas woman's mystery death wide open: Her agonizing'final gasp'... unthinkably vile corpse claims... and sick past of man who saw her last Taylor Swift's bottomless thirst for attention, her greed and sheer tackiness are now truly unbearable... this latest stunt has shown her true colors: MAUREEN CALLAHAN Spy world panic as Tulsi Gabbard prepares to unleash bombshell file dumps on secret CIA'mind control' project and Dr. Fauci Beloved mattress company backed by Travis Kelce files for bankruptcy... but makes key promise to customers Olivia Wilde, 42, complains about being on Maxim's Hot 100 List calling it the'most f***** up thing in the world' All my friends are suddenly getting divorced. Mid-life wives share taboo sex confessions about why they really leave... including common position that made one hate her husband: JANA HOCKING Kanye West's wife Bianca Censori raises eyebrows in plunging white lace lingerie as she photographs a nude model at Art Basel in Switzerland Knicks set to come face to face with Trump after president was'thunderously booed' at NBA Finals game Sensational REAL reason Jelly Roll is divorcing Bunnie XO: Insiders reveal'preacher's wife' bombshell that's the talk of Nashville... truth about legendary rocker cuckolding rumor... and G-string mishap Has Taylor Swift already revealed her wedding dress designer? Teen tourist thrown to death by Central Park horse was trying to save mom who flew out of carriage during family's first visit to Big Apple Father keeps his cool as shouting man calls cops on him for taking his two young daughters into women's restroom Trump privately frets Bibi Netanyahu's zeal to'bomb everyone' could turn him into another disgraced president'Moscow will burn', Zelensky vows as Russia's capital is blanketed in toxic smoke following huge Ukraine drone attack He drove a Rolls-Royce and lived the American dream. But behind the Gucci was the ATF's most unlikely secret weapon. What do aliens EAT? Scientist reveals the foods extraterrestrials would go for on Earth - and why E.T.'s favourite Reese's Pieces are off the cards In the 1982 blockbuster, E.T. the Extra-Terrestrial, E.T. is lured out of hiding using a trail of Reese's Pieces.

'Ringleader' of alleged UFC drone attack to kill Trump is unmasked as illegal migrant who was granted DACA stay under Obama Spy world panic as Tulsi Gabbard prepares to unleash bombshell file dumps on secret CIA'mind control' project and Dr. Fauci Watch horrifying drone video that follows woman's plunge to death after bungee team threw her from bridge without rope Horrific new videos blow Texas woman's mystery death wide open: Her agonizing'final gasp'... unthinkably vile corpse claims... and sick past of man who saw her last Taylor Swift's bottomless thirst for attention, her greed and sheer tackiness are now truly unbearable... this latest stunt has shown her true colors: MAUREEN CALLAHAN Kanye West's wife Bianca Censori raises eyebrows in plunging white lace lingerie as she photographs a nude model at Art Basel in Switzerland Trump says'fools who think I haven't been tough enough on Iran' are'jealous or stupid' after signing widely-criticised deal that includes giving Tehran $300billion Father keeps his cool as shouting man calls cops on him for taking his two young daughters into women's restroom All my friends are suddenly getting divorced. Mid-life wives share taboo sex confessions about why they really leave... including common position that made one hate her husband: JANA HOCKING Sensational REAL reason Jelly Roll is divorcing Bunnie XO: Insiders reveal'preacher's wife' bombshell that's the talk of Nashville... truth about legendary rocker cuckolding rumor... and G-string mishap Brooklyn Beckham is savaged by fans for yet another'classless' swipe at his estranged family as new DoorDash ad is branded a'giant PR mess' LIZ JONES: The cracks in Harry and Meghan's perfect facade have started to show. It's so obvious he's tiring of her tone-deaf approach... and I predict there's serious trouble in store Every emotional moment from the Gilgo Beach killer's sentencing: Rex Heuermann's shocking first words... and the chilling exchange that silenced the room Tropical Storm Arthur has formed in the Gulf, becoming the first named storm of the 2026 Atlantic hurricane season. The National Hurricane Center (NHC) announced Wednesday morning that Arthur had strengthened into a tropical storm with maximum sustained winds of 40mph. The storm was located about 40 miles northeast of Port O'Connor, Texas, and about 190 miles west-southwest of Lake Charles, Louisiana .

We develop a theoretical framework for generalization in the interpolating regime of statistical learning. The central question is why highly overparameterized estimators can attain zero empirical risk while still achieving nontrivial predictive accuracy, and how to characterize the boundary between benign and destructive overfitting. We introduce a spectral-transport stability framework in which excess risk is controlled jointly by the spectral geometry of the data distribution, the sensitivity of the learning rule under single-sample replacement, and the alignment structure of label noise. This leads to a scale-dependent Fredriksson index that combines effective dimension, transport stability, and noise alignment into a single complexity parameter for interpolating estimators. We prove finite-sample risk bounds, establish a sharp benign-overfitting criterion through the vanishing of the index along admissible spectral scales, and derive explicit phase-transition rates under polynomial spectral decay. For a model-specific specialization, we obtain an explicit theorem for polynomial-spectrum linear interpolation, together with a proof of the resulting rate. The framework also clarifies implicit regularization by showing how optimization dynamics can select interpolating solutions of minimal spectral-transport energy. These results connect algorithmic stability, double descent, benign overfitting, operator-theoretic learning theory, and implicit bias within a unified structural account of modern interpolation.

Simulation-based inference (SBI) with neural networks has accelerated and transformed cognitive modeling workflows. SBI enables modelers to fit complex models that were previously difficult or impossible to estimate, while also allowing rapid estimation across large numbers of datasets. However, the utility of SBI for iterating over varying modeling assumptions remains limited: changing parameterizations, generative functions, priors, and design variables all necessitate model retraining and hence diminish the benefits of amortization. To address these issues, we pilot a meta-amortized framework for cognitive modeling which we nickname the CogFormer. Our framework trains a transformer-based architecture that remains valid across a combinatorial number of structurally similar models, allowing for changing data types, parameters, design matrices, and sample sizes. We present promising quantitative results across families of decision-making models for binary, multi-alternative, and continuous responses. Our evaluation suggests that CogFormer can accurately estimate parameters across model families with a minimal amortization offset, making it a potentially powerful engine that catalyzes cognitive modeling workflows.

We study forward and inverse problems for time-dependent probability measures in Bayes--Hilbert spaces. On the forward side, we show that each sufficiently regular Bayes--Hilbert path admits a canonical dynamical realization: a weighted Neumann problem transforms the log-density variation into the unique gradient velocity field of minimum kinetic energy. This construction induces a transport form on Bayes--Hilbert tangent directions, which measures the dynamical cost of realizing prescribed motions, and yields a flow-matching interpretation in which the canonical velocity field is the minimum-energy execution of the prescribed path. On the inverse side, we formulate reconstruction directly on Bayes--Hilbert path space from time-dependent indirect observations. The resulting variational problem combines a data-misfit term with the transport action induced by the forward geometry. In our infinite-dimensional setting, however, this transport geometry alone does not provide sufficient compactness, so we add explicit temporal and spatial regularization to close the theory. The linearized observation operator induces a complementary observability form, which quantifies how strongly tangent directions are seen through the data. Under explicit Sobolev regularity and observability assumptions, we prove existence of minimizers, derive first-variation formulas, establish local stability of the observation map, and deduce recovery of the evolving law, its score, and its canonical velocity field under the strong topologies furnished by the compactness theory.

We study the role of batch size in stochastic conditional gradient methods under a $μ$-Kurdyka-Łojasiewicz ($μ$-KL) condition. Focusing on momentum-based stochastic conditional gradient algorithms (e.g., Scion), we derive a new analysis that explicitly captures the interaction between stepsize, batch size, and stochastic noise. Our study reveals a regime-dependent behavior: increasing the batch size initially improves optimization accuracy but, beyond a critical threshold, the benefits saturate and can eventually degrade performance under a fixed token budget. Notably, the theory predicts the magnitude of the optimal stepsize and aligns well with empirical practices observed in large-scale training. Leveraging these insights, we derive principled guidelines for selecting the batch size and stepsize, and propose an adaptive strategy that increases batch size and sequence length during training while preserving convergence guarantees. Experiments on NanoGPT are consistent with the theoretical predictions and illustrate the emergence of the predicted scaling regimes. Overall, our results provide a theoretical framework for understanding batch size scaling in stochastic conditional gradient methods and offer guidance for designing efficient training schedules in large-scale optimization.

For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a classical tube formula due to Weyl, we introduce a method to measure the decision boundary of a neural network through local surface volumes, providing a theoretically justifiable and efficient measure enabling a geometric interpretation of the effectiveness of the model applicable to the high dimensional feature spaces considered in deep learning. A smaller surface volume is expected to correspond to lower model complexity and better generalisation. We verify, on a number of image processing tasks with convolutional architectures that decision boundary volume is inversely proportional to classification accuracy. Meanwhile, the relationship between local surface volume and generalisation for fully connected architecture is observed to be less stable between tasks. Therefore, for network architectures suited to a particular data structure, we demonstrate that smoother decision boundaries lead to better performance, as our intuition would suggest.

Random sample consensus (RANSAC), which is based on a repetitive sampling from a given dataset, is one of the most popular robust estimation methods. In this study, an energy-based model (EBM) for robust estimation that has a similar scheme to RANSAC, energy-based RANSAC (EB-RANSAC), is proposed. EB-RANSAC is applicable to a wide range of estimation problems similar to RANSAC. However, unlike RANSAC, EB-RANSAC does not require a troublesome sampling procedure and has only one hyperparameter. The effectiveness of EB-RANSAC is numerically demonstrated in two applications: a linear regression and maximum likelihood estimation.