Selecting an appropriate kernel is a central challenge in kernel-based spectral methods. In \emph{Kernelized Diffusion Maps} (KDM), the kernel determines the accuracy of the RKHS estimator of a diffusion-type operator and hence the quality and stability of the recovered eigenfunctions. We introduce two complementary approaches to adaptive kernel selection for KDM. First, we develop a variational outer loop that learns continuous kernel parameters, including bandwidths and mixture weights, by differentiating through the Cholesky-reduced KDM eigenproblem with an objective combining eigenvalue maximization, subspace orthonormality, and RKHS regularization. Second, we propose an unsupervised cross-validation pipeline that selects kernel families and bandwidths using an eigenvalue-sum criterion together with random Fourier features for scalability. Both methods share a common theoretical foundation: we prove Lipschitz dependence of KDM operators on kernel weights, continuity of spectral projectors under a gap condition, a residual-control theorem certifying proximity to the target eigenspace, and exponential consistency of the cross-validation selector over a finite kernel dictionary.

Gear Computers 10 things to know about Apple's new M5 Pro and M5 Max MacBook Pros The latest versions of Apple's MacBook Pro laptops include M5 chips with revamped architecture to bring performance upgrades across the board. The new computers look similar, but the guts have gotten revamped. We may earn revenue from the products available on this page and participate in affiliate programs. Apple's latest MacBook Pro refresh landed today with two new processors, the M5 Pro and M5 Max, built on what the company calls its Fusion Architecture. We have already been using the vanilla M5 chip in the latest version of the Apple Vision Pro headset, but these new MBP models crank up the power level even more.

Why it's high time we stopped anthropomorphising ants We have long drawn parallels between ants and humans. Now we are comparing the insects to computers. Pollution is making many cities unlivable for their human inhabitants, but it is also tearing ant families and communities apart. Ants recognise each other by sniffing a thin layer of hydrocarbons on the outside of their exoskeletons; each colony has a specific "smell". But a new study reveals that ozone emissions can change the structure of these hydrocarbons.

The fundamental problem of non-parametric two-sample testing consists in detecting the difference between any two distributions having access only to samples from these.

Points were sampled uniformly in the bands denoted by dashed lines. We posit that these barriers are due, at least in part, to the sensitivity of decision trees to transformations of the input resulting from greedy construction and simple decision rules. Of these, key limitation is the latter; even if we replace greedy construction with a perfect tree learner, simple distributions can nonetheless require an arbitrarily large axis-aligned tree to fit.

We present a novel multi-agent RL approach, Selective Multi-Agent Prioritized Experience Relay, in which agents share with other agents a limited number of transitions they observe during training.

Generalizable implicit neural representation (INR) enables a single continuous function, i.e., a coordinate-based neural network, to represent multiple data instances

But, they are limited to unimodal, Gaussian approximations of state uncertainty.