Fraud detection in payment networks relies on labels generated through heterogeneous and imperfect observation processes, yet existing approaches treat fraud as a homogeneous binary variable. We show that this assumption is structurally incorrect and leads to provable inefficiency. We introduce an observation-mechanism taxonomy that partitions fraud into five classes, each defined by a distinct censorship and labeling pipeline. We prove that estimating fraud rates separately by class and aggregating strictly dominates pooled estimation, with the efficiency gap characterized as a Jensen penalty arising from heterogeneous observation rates. For each class, we derive the binding theoretical constraint on detection, including endogenous label corruption, structural non-observability, and feature non-informativeness. These results establish that fraud detection is fundamentally a collection of distinct estimation problems, each governed by its own observation structure and detection limit.

This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate $\mathcal{O}((WL)^{-2\tilde{s}})$ for anisotropic Besov space $\mathcal{B}^{\boldsymbol{s}}_{q,r}([0,1]^d)$ with anisotropic smoothness $\boldsymbol{s}=(s_1,\dots,s_d)$ under the embedding condition $\tilde{s} > 1/q-1/p$, where the mean smoothness $\tilde{s} = (\sum_{i=1}^d s_i^{-1})^{-1}$. For mixed smooth Besov space $\mathcal{MB}^s_{q,r}([0,1]^d)$ with mixed smoothness $s>1/q-1/p$, we show that the approximation rate $\mathcal{O}((WL)^{-2s})$ holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

MSI's Claw 8 EX AI+ handheld comes out on June 23 MSI's Claw 8 EX AI+ handheld comes out on June 23 The premium gaming handheld's price is expected to be around $1,500. A new wave of premium handhelds is on the way and MSI's upcoming Claw 8 EX AI+ is leading the charge. The company revealed its latest handheld during a hands-on event at its headquarters in Taipei, Taiwan before Computex 2026. The Claw 8 EX AI+ will follow up MSI's previous handhelds introduced in 2024 that start at around $800, but is reportedly going to cost as much as $1,500 . It comes out on June 23.

Two lawmakers in the United States Congress are pushing to repeal a section in an upcoming defence budget that would deepen integration between the US and Israeli militaries. Their effort comes amid outcry against the latest draft of the annual National Defense Authorization Act (NDAA), a bill that sets the budget for the US military. On Sunday, Democratic Congressman Ro Khanna said he would use his seat on the House Armed Services Committee to put forward an amendment to remove Section 224 from the bill. His Republican colleague Thomas Massie also warned that, if the bill is approved by the committee, he would work to oppose it on the floor of the House of Representatives. He framed the issue as a question of US independence from foreign governments.

Why Iran won't give up Hormuz Could Israel sabotage US-Iran deal? Lawmakers in the United States are quietly advancing a proposal that could deepen military ties between the US and Israel in unprecedented ways, at a time when public support for Israel among Americans is increasingly fractured. Among the provisions included in the 2027 National Defence Authorisation Act (NDAA) released this week is Section 224, the "United States-Israel Defence Technology Cooperation Initiative". Some legislators have already signalled opposition, with Representative Thomas Massie saying he would seek to remove the provision if it reaches the House floor. The measure remains at an early stage, but analysts say if passed, it would limit political oversight over the defence relationship.

Cardinals fan tries to throw Pete Crow-Armstrong's home run ball back onto the field and fails miserably National anthem'standoff' between Rays and Angels players leads to pregame ejections at Tropicana Field Caitlin Clark's turnover problem could be a defining factor in her legendary WNBA career Donald Trump shouts out Jaxson Dart after Giants QB's appearance at event was met with controversy Jaxson Dart shouldn't have to justify introducing Trump at a rally, but this is what he should have said The 2007 Mountaineers remain college football's greatest'what-if' story nearly two decades later AB Hernandez advances in California state championship as Save Girls' Sports activists rally nearby Tennis player Rafael Jodar accused of pushing French Open ball girl, but did he really? Rebecca Grant: Trump is'squeezing Iran like a vice' to force nuclear compliance Dan Hoffman weighs in on'horrific' ex-CIA official gold bar scandal Former Jill Biden spokesperson says memoir aims to'change the tape' Florida Gov DeSantis addresses wealth migration to Florida, points to'destructive policies' elsewhere Chad Bianco blasts GOP voices for urging him to drop out of California governor's race Democrat responds to fallout from Jill Biden's comments on 2024 debate The two-time U.S. Open champion discussed torque and grip pressure with Google's Gemini after a 1-over third round Pro-golfer Bryson DeChambeau detailed his vision for a family during an appearance on The Katie Miller Podcast on Tuesday, revealing he wants four kids. Bryson DeChambeau may have finished third in LIV Golf's latest event in South Korea, but that doesn't mean he's pleased with his game. DeChambeau, who won back-to-back LIV events in March, found himself one shot back of the lead after the opening two rounds in South Korea, but put together a disappointing third round at 1-over par. As is often the case for DeChambeau after what he deems to be a lackluster performance, it turned into a long night on the driving range searching for something.

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From digital twins to models'sculpted' by programmers, generative AI has been popping up all over the fashion industry. When an Australian e-commerce retailer started using AI-generated models to sell products, lifestyle editor Alyx Gorman had to see if the garments were more than mere pixels. The Iconic, which sells the dress worn in this video, said in a statement: 'Where AI-generated imagery is used to advertise products for sale on our platform, our expectation is that it is clearly labelled and that the product itself is represented as accurately as possible for customers.' Meanwhile, Atoir, the designer, said: 'The Australian fashion industry is highly competitive, particularly for independent brands. We believe that when used responsibly, tools like this can help smaller businesses to operate with greater agility while still maintaining the creative standards and product integrity that matter to both the brand and the customer.'

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Pope Leo's Magnifica Humanitas encyclical addresses AI, Catholic social teaching, war, and peace, but its master motif is the irreducible dignity of the human being.