This paper studies robust nonparametric regression, in which an adversarial attacker can modify the values of up to $q$ samples from a training dataset of size $N$. Our initial solution is an M-estimator based on Huber loss minimization. Compared with simple kernel regression, i.e. the Nadaraya-Watson estimator, this method can significantly weaken the impact of malicious samples on the regression performance. We provide the convergence rate as well as the corresponding minimax lower bound. The result shows that, with proper bandwidth selection, $\ell_\infty$ error is minimax optimal. The $\ell_2$ error is optimal with relatively small $q$, but is suboptimal with larger $q$. The reason is that this estimator is vulnerable if there are many attacked samples concentrating in a small region. To address this issue, we propose a correction method by projecting the initial estimate to the space of Lipschitz functions. The final estimate is nearly minimax optimal for arbitrary $q$, up to a $\ln N$ factor.

Removal or cancellation of noise has wide-spread applications for imaging and acoustics. In every-day-life applications, denoising may even include generative aspects, which are unfaithful to the ground truth. For scientific use, however, denoising must reproduce the ground truth accurately. Here, we show how data can be denoised via a deep convolutional neural network such that weak signals appear with quantitative accuracy. In particular, we study X-ray diffraction on crystalline materials. We demonstrate that weak signals stemming from charge ordering, insignificant in the noisy data, become visible and accurate in the denoised data. This success is enabled by supervised training of a deep neural network with pairs of measured low- and high-noise data. We demonstrate that using artificial noise does not yield such quantitatively accurate results. Our approach thus illustrates a practical strategy for noise filtering that can be applied to challenging acquisition problems.

Causal representation learning has emerged as the center of action in causal machine learning research. In particular, multi-domain datasets present a natural opportunity for showcasing the advantages of causal representation learning over standard unsupervised representation learning. While recent works have taken crucial steps towards learning causal representations, they often lack applicability to multi-domain datasets due to over-simplifying assumptions about the data; e.g. each domain comes from a different single-node perfect intervention. In this work, we relax these assumptions and capitalize on the following observation: there often exists a subset of latents whose certain distributional properties (e.g., support, variance) remain stable across domains; this property holds when, for example, each domain comes from a multi-node imperfect intervention. Leveraging this observation, we show that autoencoders that incorporate such invariances can provably identify the stable set of latents from the rest across different settings.

While there has been a lot of progress in developing deep neural networks (DNNs) to solve practical machine learning problems [1-3], the inner workings of neural networks is not well understood. A foundational theory for understanding how neural networks work is still lacking despite extensive research over several decades. In this paper, we provide a novel analysis of neural networks based on geometric algebra and convex optimization. We show that weights of deep ReLU neural networks learn the wedge product of a subset of training samples when trained by minimizing standard regularized loss functions. Furthermore, the training problem reduces to convex optimization over wedge product features, which encode the geometric structure of the training dataset. This structure is given in terms of signed volumes of triangles and parallelotopes generated by data vectors. Our analysis provides a novel perspective on the inner workings of deep neural networks and sheds light on the role of the hidden layers.

Chairman of the Joint Chiefs of Staff Gen. Charles Q. Brown Jr. joins'Fox News Sunday' to discuss a new survey that revealed 74% of Americans are concerned about a war between the U.S. and China. This is a rush transcript of'Fox News Sunday' on December 3, 2023. This copy may not be in its final form and may be updated. A special hour on the state of defense, a report card on America's military readiness to meet the challenges of an increasingly dangerous world. Israel's war with Hamas, the latest conflict to ignite instability, turbo- charging attacks on our forces in the region from Iranian proxies. We'll get reaction from National Security Council Communications Coordinator John Kirby about the restart of the war and the headwinds the Biden White House faces from Democrats over conditioning future aid to Israel. GENERAL C.Q. BROWN, JOINT CHIEFS CHAIRMAN: We want to be so good at what we do that our adversaries go, not today, not tomorrow, not ever. General C.Q. Brown joins me here at the Reagan Library. And before serving in Congress, they served several tours of duty on the ground in two of America's longest wars. We sit down with Congressman Michael Waltz and Seth Moulton, veterans for both sides of the aisle, as the fight over defense spending is coming up against the stark deadline. Plus -- JENNIFER GRIFFIN, FOX NEWS NATIONAL SECURITY CORRESPONDENT: Is it cool to be patriotic now? UNIDENTIFIED MALE: It's always been cool to be contrarian and I think right now, it's -- it's been a little contrarian to be very patriotic. BREAM: Our inside look at how cutting-edge technology is shaping the future of warfare and battlefields worldwide. Here are the top headlines making news today. Israel is widening its evacuation orders for Palestinians in southern Gaza, including in and around the cities of Khan Younis and Rafah, which both reported heavy bombardment overnight. Israeli Prime Minister Benjamin Netanyahu calling for a total victory against Hamas and pushing back against White House calls to allow the Palestinian Authority to ultimately govern Gaza, claiming the group also calls for Israel's destruction. Meanwhile, in Paris, French authorities are looking into whether terrorism was to blame for a knife and hammer attack on tourists near the Eiffel Tower, leaving a German man dead and two others injured. A 26-year-old French national has been arrested. Let's turn now to Trey Yingst in southern Israel with the very latest on the war in Gaza. After a week-long ceasefire saw more than 100 hostages freed from Gaza, fighting has resumed for a third day. Israeli officials say the ground and air campaign in the second phase of this war against the strip could last for months. New airstrikes overnight targeted tunnel shafts and weapon storage facilities.

The string of Autopilot crashes reveals the consequences of allowing a rapidly evolving technology to operate on the nation's roadways without significant government oversight, experts say. While NHTSA has several ongoing investigations into the company and specific crashes, critics argue the agency's approach is too reactive and has allowed a flawed technology to put Tesla drivers -- and those around them -- at risk. The approach contrasts with federal regulation of planes and railroads, where crashes involving new technology or equipment -- such as recurring issues with Boeing's 737 Max -- have resulted in sweeping action by agencies or Congress to ground planes or mandate new safety systems. Unlike planes, which are certified for airworthiness through a process called "type certification," passenger car models are not prescreened, but are subject to a set of regulations called Federal Motor Vehicle Safety Standards, which manufacturers face the burden to meet.

The recently-departed watchdog in charge of monitoring facial recognition technology has joined the private firm he controversially approved, paving the way for the mass roll-out of biometric surveillance cameras in high streets across the country. In a move critics have dubbed an "outrageous conflict of interest", Professor Fraser Sampson, former biometrics and surveillance camera commissioner, has joined Facewatch as a non-executive director. Sampson left his watchdog role on 31 October, with Companies House records showing he was registered as a company director at Facewatch the following day, 1 November. Campaigners claim this might mean he was negotiating his Facewatch contract while in post, and have urged the advisory committee on business appointments to investigate if it may have "compromised his work in public office". It is understood that the committee is currently considering the issue.

A French frigate has shot down two drones over the Red Sea that were believed to be approaching from the coast of Yemen, according to the French military. "The interception and destruction of these two identified threats" were carried out late on Saturday by the frigate Languedoc, which operates in the Red Sea, the general staff said in a press release on Sunday. The interceptions happened at 20:30 GMT and 22:30 GMT and were 110km (68 miles) from the Yemeni coast, it added. Yemen's Iran-backed Houthi rebels on Saturday threatened to attack any vessels heading to Israeli ports unless food and medicine were allowed into the besieged Gaza Strip, which has been devastated by more than two months of Israeli bombing. The Houthis said that all "ships linked to Israel or that will transport goods to Israeli ports" are not welcome in the Red Sea, a vital channel for global trade linked to the Suez Canal.

Machine learning (ML) is emerging as a transformative tool for the design of architected materials, offering properties that far surpass those achievable through lab-based trial-and-error methods. However, a major challenge in current inverse design strategies is their reliance on extensive computational and/or experimental datasets, which becomes particularly problematic for designing micro-scale stochastic architected materials that exhibit nonlinear mechanical behaviors. Here, we introduce a new end-to-end scientific ML framework, leveraging deep neural operators (DeepONet), to directly learn the relationship between the complete microstructure and mechanical response of architected metamaterials from sparse but high-quality in situ experimental data. The approach facilitates the inverse design of structures tailored to specific nonlinear mechanical behaviors. Results obtained from spinodal microstructures, printed using two-photon lithography, reveal that the prediction error for mechanical responses is within a range of 5 - 10%. Our work underscores that by employing neural operators with advanced micro-mechanics experimental techniques, the design of complex micro-architected materials with desired properties becomes feasible, even in scenarios constrained by data scarcity. Our work marks a significant advancement in the field of materials-by-design, potentially heralding a new era in the discovery and development of next-generation metamaterials with unparalleled mechanical characteristics derived directly from experimental insights.

Adversarial training is one of the most popular methods for training methods robust to adversarial attacks, however, it is not well-understood from a theoretical perspective. We prove and existence, regularity, and minimax theorems for adversarial surrogate risks. Our results explain some empirical observations on adversarial robustness from prior work and suggest new directions in algorithm development. Furthermore, our results extend previously known existence and minimax theorems for the adversarial classification risk to surrogate risks.