We study the population loss landscape of two-layer ReLU networks of the form $\sum_{k=1}^K \mathrm{ReLU}(w_k^\top x)$ in a realisable teacher-student setting with Gaussian covariates. We show that local minima admit an exact low-dimensional representation in terms of summary statistics, yielding a sharp and interpretable characterisation of the landscape. We further establish a direct link with one-pass SGD: local minima correspond to attractive fixed points of the dynamics in summary statistics space. This perspective reveals a hierarchical structure of minima: they are typically isolated in the well-specified regime, but become connected by flat directions as network width increases. In this overparameterised regime, global minima become increasingly accessible, attracting the dynamics and reducing convergence to spurious solutions. Overall, our results reveal intrinsic limitations of common simplifying assumptions, which may miss essential features of the loss landscape even in minimal neural network models.

Accurate representation of non-Gaussian distributions of quantities of interest in nonlinear dynamical systems is critical for estimation, control, and decision-making, but can be challenging when forward propagations are expensive to carry out. This paper presents an approach for estimating probability density functions of states evolving under nonlinear dynamics using Seminonparametric (SNP), or Gallant-Nychka, densities. SNP densities employ a probabilists' Hermite polynomial basis to model non-Gaussian behavior and are positive everywhere on the support by construction. We use Monte Carlo to approximate the expectation integrals that arise in the maximum likelihood estimation of SNP coefficients, and introduce a convex relaxation to generate effective initial estimates. The method is demonstrated on density and quantile estimation for the chaotic Lorenz system. The results demonstrate that the proposed method can accurately capture non-Gaussian density structure and compute quantiles using significantly fewer samples than raw Monte Carlo sampling.

Stretches, exercises, and myofascial options to help your achy feet. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. First invest in supportive shoes. Breakthroughs, discoveries, and DIY tips sent six days a week. Anecdotally associated with the obese and pregnant, it's foot pain and tightness common among all walks of adult, particularly those between the ages of 40 and 60 . But it's important to know that you're unlikely to take a wrong step and suddenly catch plantar fasciitis.

Staff at a wildlife center paired a baby-less owl mother with mother-less baby owl. More information Adding us as a Preferred Source in Google by using this link indicates that you would like to see more of our content in Google News results. The mama owl ended up adopting another owlet too. Breakthroughs, discoveries, and DIY tips sent six days a week. When animals end up in a wildlife center, it's usually not for happy reasons.

This adds another way for your inner life to be read by prying eyes, though you can opt out. A lifelong diary writer himself, Reinberg launched the app because he was fascinated by journalling, psychology and tech. He has no professional background or education in therapy. "We are not a clinical or a therapy tool," he says. "We're focused on self-reflection and finding connections between entries, holding up a mirror that helps you to make progress in your life." One feature I don't like is that it analyses each entry and gives a percentage score for your dominant emotions.

OpenAI logo is seen in this illustration taken on 20 May 2024. OpenAI logo is seen in this illustration taken on 20 May 2024. AI companies know they have an image problem. OpenAI made a surprise announcement this week - not an update to ChatGPT or another multibillion-dollar datacenter - but a policy paper that called for a reimagining of the social contract based around "a slate of people-first ideas". It's the latest move in an aggressive effort by the major AI players to reshape the narrative around their industry, as polls show public disapproval of AI increasing.

'Releasing a marketing post with purposely vague language that obscures evidence brings into question if they are trying to garner further investment without scrutiny,' one scientist said. 'Releasing a marketing post with purposely vague language that obscures evidence brings into question if they are trying to garner further investment without scrutiny,' one scientist said. 'Too powerful for the public': inside Anthropic's bid to win the AI publicity war This week, the AI company Anthropic said it had created an AI model so powerful that, out of a sense of overwhelming responsibility, it was not going to release it to the public. The US treasury secretary, Scott Bessent, summoned the heads of major banks for a chat about the model, Mythos. The Reform UK MP Danny Kruger wrote a letter to the government urging it to " engage with AI firm Anthropic whose new frontier model Claude Mythos could present catastrophic cybersecurity risks to the UK".

Ritzy Bay Area town torn apart after teacher's daughter, 16, was behind wheel when four friends died in high-speed crash... then she posted a TikTok video that poured fuel on the flames Two CIA officers killed in Mexico when their car skidded off ravine and exploded after meeting about bust of'largest ever drug lab' Insiders claim failed AI rollout could be to blame for Tim Cook's departure from Apple - as one says'the AI era requires a different kind of leadership' Trump confronts Xi as US forces seize Chinese ship carrying mysterious'gift' to Iran New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-ageing, shrinks pores, smooths wrinkles... and even banishes rosacea Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' Fury as murderer marries pen pal behind bars... as teenage victim's mom says: 'I'm serving a life sentence without my son' Kate and William join Charles and Camilla in celebrating British centenarians at Buckingham Palace as Royal Family marks the late Queen's 100th birthday US troops board second tanker as Trump accuses Iran of violating ceasefire'numerous times' - Live updates AMANDA PLATELL: Why Sarah Ferguson - with the ghost of Princess Diana at her side - is ready to sensationally blow up the Royal Family. She knows ALL their secrets... New Jersey man's chilling'cancer map' fuels fears of poisoned neighborhood with 41 cases and counting How to lose weight when perimenopause sabotages your metabolism: I'm a trainer but when I hit 46, I piled on the pounds overnight. I was losing hair so fast a bald spot the size of an orange appeared. I owe my life to a $1 at-home treatment that REVERSED the damage in a month.

Ritzy Bay Area town torn apart after teacher's daughter, 16, was behind wheel when four friends died in high-speed crash... then she posted a TikTok video that poured fuel on the flames Two CIA officers killed in Mexico when their car skidded off ravine and exploded after meeting about bust of'largest ever drug lab' Nancy Guthrie sheriff's appalling past revealed: Beat handcuffed suspect so badly he needed intensive care, used VILE language about woman and lied in sworn statement Trump confronts Xi as US forces seize Chinese ship carrying mysterious'gift' to Iran New'Hollywood dose' pill: A-listers hooked on'youth elixir' that dermatologists say is anti-ageing, shrinks pores, smooths wrinkles... and even banishes rosacea Days after we got engaged, the love of my life told me he'd killed a man and buried him in a bog. I reported him to police... but then I made this irreversible mistake Ark of the Covenant's final resting place pinpointed by archaeologists as fresh search begins Fury as murderer marries pen pal behind bars... as teenage victim's mom says: 'I'm serving a life sentence without my son' Insiders claim failed AI rollout could be to blame for Tim Cook's departure from Apple - as one says'the AI era requires a different kind of leadership' Life-threatening cantaloupe recall in four states upgraded to FDA's highest risk level... 'reasonable probability of death' AMANDA PLATELL: Why Sarah Ferguson - with the ghost of Princess Diana at her side - is ready to sensationally blow up the Royal Family. She knows ALL their secrets... Team USA Olympics star Noah Lyles slammed for'horrible' reaction to his wife's wedding dress reveal In honour of the Queen's (purple!) reign: Kate mirrors late monarch's colourful wardrobe and wears her pearl earrings and necklace US troops board second tanker as Iran is accused of breaking ceasefire'numerous times' How to lose weight when perimenopause sabotages your metabolism: I'm a trainer but when I hit 46, I piled on the pounds overnight. The new'posh' drug that's easier to order than Uber Eats - and why all my middle-class friends have ditched booze and cocaine for it: JANA HOCKING Autistic woman, 24, worked hard to build independent life for herself... now she's PARALYZED thanks to selfishness of stranger The video game where you play as JESUS: Open-world simulator lets you'follow the path' of the Messiah as he is baptized, perform miracles, fights the Devil and gets CRUCIFIED READ MORE: Rare marble artefact discovered in the'cradle of Christianity' While fans eagerly await the release of Grand Theft Auto 6, one Polish game studio has unveiled a rather unorthodox alternative. The video game, titled'I Am Jesus Christ' allows players to'walk in the footsteps of Jesus' in an immersive first-person retelling of the gospel.

'Lethal cyber-attacks are thankfully rare. But a new AI release could change that.' 'Lethal cyber-attacks are thankfully rare. But a new AI release could change that.' Anthropic's new AI tool has implications for us all - whether we can use it or not Claude Mythos's apparent superhuman hacking abilities are alarming experts as the Trump administration remains blinded by hostility I n June 2024, a cyber-attack on a pathology services company caused chaos across London's hospitals. More than 10,000 appointments were cancelled. Blood shortages followed and delays to blood tests led to a patient's death . Lethal cyber-attacks like this are thankfully rare.