'Record the faces': Tense moment NBA boss gives VERY honest take on Trump attending Knicks game Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Disgraceful texts'hot' teacher sent boy, 17, who she had illegal sex with where she moaned about her HUSBAND Everyone always said I cleared my throat a lot. But then I developed shoulder pain and doctors discovered the sinister cause... the world's deadliest cancer. Don't leave it too late like I did Outrage as Netanyahu is caught SPYING on Trump's Iran negotiators... as JD Vance reveals a chilling truth about Israel White couple gave birth to'non-Caucasian' baby. Parents were told son, 7, had ADHD... not realizing he was battling terrifying disease that has now left him BLIND'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege Medical student, 24, died by suicide in his white coat a day after he was suspended for alleged'inappropriate' behavior towards female patient, lawsuit alleges, as his heartbreaking goodbye note to parents is revealed Karmelo Anthony's parents seen leaving the courtroom in tears just before son's defense team pulls shock move Grim-faced former Louisiana mayor Misty Roberts arrives in court for sentencing after being found guilty of having sex with son's teenage friend Mother died during tummy tuck and Brazilian butt lift after clinic staff failed to hold'slow' elevator for EMTs, report alleges Gaming influencer Alex Cimo dies'very suddenly' aged 32 just a month after'refusing to accept his fate' The porn-fuelled fantasy middle-class husbands are desperate to try with their wives... and it almost always ends in divorce: JANA HOCKING All the backstage gossip from Miami Swim Week: Insider exposes'catty' VIP's diva demands... STEALING... and'morbidly embarrassing' celeb moment everyone is whispering about Girl, 13, mistakenly told she was DYING after Oregon hospital staff made jaw-dropping surgical mistake, parents' $17m lawsuit alleges Mother's final words before she was shot dead'by new husband' in front of her two young children'They have a problem with my country': Africa's best referee, who was denied entry to the US and will miss the World Cup, speaks out and insists he had a valid visa Furious dad films his partner in bed with his 19-year-old son: You've seen the viral video - now all three tell the Daily Mail what REALLY happened in the scandal gripping Australia World's shark attack hotspots revealed: As a great white is spotted in the Mediterranean, experts reveal the areas where you're most likely to be bitten The world's shark attack hotspots have been revealed, after a g reat white shark was spotted in the Mediterranean Sea. The enormous predator was recorded between Sicily and Tunisia, in what is believed to be the first ever footage captured of an adult great white in the area.

'Record the faces': Tense moment NBA boss gives VERY honest take on Trump attending Knicks game Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Disgraceful texts'hot' teacher sent boy, 17, who she had illegal sex with where she moaned about her HUSBAND Everyone always said I cleared my throat a lot. But then I developed shoulder pain and doctors discovered the sinister cause... the world's deadliest cancer. Don't leave it too late like I did Outrage as Netanyahu is caught SPYING on Trump's Iran negotiators... as JD Vance reveals a chilling truth about Israel White couple gave birth to'non-Caucasian' baby. Parents were told son, 7, had ADHD... not realizing he was battling terrifying disease that has now left him BLIND Medical student, 24, died by suicide in his white coat a day after he was suspended for alleged'inappropriate' behavior towards female patient, lawsuit alleges, as his heartbreaking goodbye note to parents is revealed Karmelo Anthony's parents seen leaving the courtroom in tears just before son's defense team pulls shock move Grim-faced former Louisiana mayor Misty Roberts arrives in court for sentencing after being found guilty of having sex with son's teenage friend Mother died during tummy tuck and Brazilian butt lift after clinic staff failed to hold'slow' elevator for EMTs, report alleges Gaming influencer Alex Cimo dies'very suddenly' aged 32 just a month after'refusing to accept his fate' 'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege The porn-fuelled fantasy middle-class husbands are desperate to try with their wives... and it almost always ends in divorce: JANA HOCKING Meghan Markle's As Ever website has had'less than 400,000 US visitors' since January - as Duchess launches collaboration with a lifestyle influencer to plug her products Nashville's most-hated influencer sparked outrage with sick posts about teen girl who vanished into the woods after a party... now his incredible life of luxury is unraveling Girl, 13, mistakenly told she was DYING after Oregon hospital staff made jaw-dropping surgical mistake, parents' $17m lawsuit alleges Mother's final words before she was shot dead'by new husband' in front of her two young children'They have a problem with my country': Africa's best referee, who was denied entry to the US and will miss the World Cup, speaks out and insists he had a valid visa Massive twist in JPMorgan'sex slave' case as accuser unveils NEW dossier of wild claims: 'The story is about to change dramatically' A great white shark has been spotted underwater in the Mediterranean for the first time ever. Divers from Healthy Seas were removing ghost nets on an offshore shipwreck between Sicily and Tunisia when they spotted the predator.

Neural networks are known to develop latent representations that are $aligned$, namely structurally similar across networks trained with different architectures, training protocols, or training datasets. We study this phenomenon in a controlled setting, where we train an ensemble of networks on regression and classification tasks using training sets perturbed by independent realizations of a noise process. We show that the signal-to-noise ratio (SNR) and the training sample size influence the alignment in qualitatively similar ways in networks trained on real-world datasets and in an extremely simple $linear$ network with a single hidden layer, for which the alignment can be estimated analytically. Across linear and nonlinear networks, regression and classification tasks, and both synthetic and real-world data, we consistently observe that alignment varies monotonically with SNR but non-monotonically with training sample size. In particular, the alignment is minimized near the interpolation threshold, and a stronger alignment does not necessarily correspond to better generalization error. These findings reveal a non-trivial dependence of alignment on data quality and quantity, decoupled from generalization performance.

The main XAI attribution methods for deep neural networks -- GradCAM, SHAP, LIME, Integrated Gradients -- operate on separate theoretical foundations and are not formally comparable. We present GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), a mathematical framework establishing a representation theory for attributions: every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM, but excludes nonlinear functionals such as standard GradCAM or attention maps. Seven formal theorems provide simultaneous guarantees absent in any individual method: (T1) necessary canonical form; (T2) exact completeness; (T3) Monte Carlo convergence O(1/sqrt(m))+O(1/k); (T4) exact Shapley Interaction Values; (T5) Hoeffding ANOVA decomposition; (T6) Sobol sensitivity generalization; (T7) multi-scale extension (MS-GRALIS) with minimum-variance weights. An algebraic appendix justifies the GRALIS-SIV correspondence via the Mobius transform without circularity. GRALIS satisfies 13.5/14 axiomatic properties vs. 2.5-6/14 for individual methods, including completeness, sensitivity, locality, order-k interactions and optimal multi-scale aggregation simultaneously. Preliminary validation on BreaKHis (1,187 histology images, DenseNet-121) reports deletion faithfulness AUC +0.015 (malignant), 96% class-conditional consistency, SAL = 0.762+/-0.109 and sparsity index 0.39. Extended comparison with baseline XAI methods is planned for a companion paper.

Heavy-tailed distributions are prevalent in performance evaluation, network traffic, and risk modeling. This behavior poses a fundamental challenge for modern deep generative models. Standard Variational Autoencoders (VAEs) employ Gaussian decoder likelihoods and Lipschitz-constrained neural networks, a combination that is structurally incapable of producing heavy-tailed outputs: the Gaussian tail decays exponentially, and Lipschitz continuity prevents the decoder from amplifying rare events from the latent space input to sufficiently overcome this decay. We provide both a theoretical characterization of this limitation and a controlled empirical demonstration using synthetic Pareto data across a grid of tail indices $α$ $\in$ {2, 3, 5, 30} and dimensions d $\in$ {1, 5, 10}. As a solution, we replace the Gaussian decoder with a Phase-Type (PH) distribution based on Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions allow for arbitrarily precise approximations of any positive-valued distributions, including heavy-tailed families. Experiments showed that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to x6 and extreme quantile error by up to x10 compared to the Gaussian baseline for heavy-tailed data. These results demonstrate that integrating Markov chain-based distributions into the decoder of a generative model institutes a principled and practically effective solution to the heavy-tail generation problem.

Marco Rubio warns China of'repercussions' as he reveals what really happened during closed-door Trump and Xi meeting Ex-Yankees star Carl Pavano'peed in shampoo bottles and soiled the bed,' ex-wife claims as bitter prenup feud takes disgusting twist Fury as Kash Patel SNORKELS at sacred war tomb where 900 sailors still lie... then jets off to Las Vegas Glamorous Texas Democrat's secret KINK exposed: Congressional candidate's past life returns to haunt her After theater groping shame, Lauren Boebert is being bankrolled by America's cringiest ex-congressman... and it exposes a MASSIVE hypocrisy Horrifying final days of killer dad Chris Watts' pregnant wife before she was slaughtered alongside their daughters. Read all the chilling texts and receipts in full for first time: 'My eyes burn from crying' RHOBH star Diana Jenkins denies claims she put Hayden Panettiere in bed with'undressed man' when she was 18 Trump reveals Xi's offer to break Iran's Hormuz chokehold... as China's price for the rescue looms Mystery blonde Trump aide with unfettered access to President's phone sparks White House friction: Real reason his posts contain random capital letters... and shadowy team behind them unmasked Despicable crimes paid for couple's lavish lifestyle that they flaunted online while gold chain-wearing husband fleeced $1BILLION from taxpayers New DNA analysis of Christopher Columbus reveals truth about explorer's origins that rewrites history Bitter cat fight erupts over DHS'sugar baby' scandal: Veteran female intelligence officer launches explosive new accusations that go right to top of counterterror HQ I lost 9lb in two weeks by making one simple tweak to my lifestyle. I didn't use Mounjaro, diet or change how I exercise and I couldn't believe the results... anyone can do it too I'm godfather to Candace Owens' daughter and Charlie Kirk was my friend... so I know the real reason she's attacking Erika - and I'll never publicly condemn her Britney Spears seen'barking and carrying knife' during chaotic restaurant visit I've had acid reflux all my life. Target customers threaten to boycott store after controversial'upgrade' to shopping cart New DNA analysis of Christopher Columbus reveals truth about explorer's origins that rewrites history A new DNA analysis of remains belonging to several direct descendants of Christopher Columbus may have uncovered a history-changing truth about the explorer's origins. For centuries, historians have believed the explorer was born in Genoa, Italy, rising from humble beginnings to persuade the Catholic Monarchs to finance what many considered an impossible voyage across the Atlantic.

Batch normalization (BN) is central to modern deep networks, but its effect on the realized function during training remains less understood than its optimization benefits. We study training-time BN in continuous piecewise-affine (CPA) networks through the geometry of switching hyperplanes and the induced affine-region partition. Conditioned on a mini-batch, we show that BN defines for each neuron a reference hyperplane through the batch centroid, and that breakpoint-switching hyperplanes are parallel translates whose offsets are expressed in batch-standardized coordinates and are independent of the raw bias. This yields an exact criterion for when a switching hyperplane intersects a local $\ell_\infty$ window and motivates a local region-density functional based on exact affine-region counts. Under explicit sufficient conditions, we show that BN increases expected local partition refinement in ReLU and more general piecewise-affine networks, and that this mechanism transfers locally through depth inside parent affine regions where the upstream representation map is an affine embedding. These results provide a function-level geometric account of training-time BN as a batch-conditional recentering mechanism near the data.

We study optimal policy learning under combined budget and minimum coverage constraints. We show that the problem admits a knapsack-type structure and that the optimal policy can be characterized by an affine threshold rule involving both budget and coverage shadow prices. We establish that the linear programming relaxation of the combinatorial solution has an O(1) integrality gap, implying asymptotic equivalence with the optimal discrete allocation. Building on this result, we analyze two implementable approaches: a Greedy-Lagrangian (GLC) and a rank-and-cut (RC) algorithm. We show that the GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples. By contrast, RC is approximately optimal whenever the coverage constraint is slack or costs are homogeneous, while misallocation arises only when cost heterogeneity interacts with a binding coverage constraint. Monte Carlo evidence supports these findings.

Link prediction (inferring missing or future connections between nodes in a graph) is a fundamental problem in network science with widespread applications in, e.g., biological systems, recommender systems, finance and cybersecurity. The ability to accurately predict links has significant real-world applications, such as detecting fraudulent financial transactions or identifying drug-target interactions in biomedicine. Despite a rich literature, link prediction is still challenging, especially for graphs enriched with information on edges (direction) and nodes (attributes). In fact, research on link prediction, especially the one based on Graph Deep Learning (GDL), has mostly focused on undirected graphs, without fully leveraging node attributes. Here, we fill this gap by proposing Gravity-GraphSAGE (GG-SAGE), a modified version of GraphSAGE, a GDL model for node embeddings, composed of a gravity-inspired decoder. This implementation is the first example in the literature of a GraphSAGE backbone adopted for directed link prediction. Using the benchmark datasets Cora, Citeseer, PubMed and 16 real-world graphs from the online Netzschleuder repository, we show that our proposed model outperforms state-of-the-art GDL link prediction techniques. Using further experimental evidence, we relate the quality of the output of our model with various characteristics of the graph, suggesting that our framework scales well when applied to data of increasing complexity.

Real-world datasets are inherently heterogeneous, yet how per-class structural differences and sampling imbalance shape the training dynamics of diffusion models-and potentially exacerbate disparities-remains poorly understood. While models typically transition from an initial phase of generalization to memorizing the training set, existing theory assumes homogeneous data, leaving open how class imbalance and heterogeneity reshape these dynamics. In this work, we develop a high-dimensional analytical framework to study class-dependent learning in score-based diffusion models. Analyzing a random-features model trained on Gaussian mixtures, we derive the feature-covariance spectrum to characterize per-class generalization and memorization times. We reveal the explicit hierarchy governing these dynamics: class variance is the primary determinant of learning order-consistently favoring higher-variance classes-while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion. Together, these results suggest that diffusion models can memorize some classes while others remain insufficiently learned. We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.