We address the problem of designing a sublinear-time spectral clustering oracle for graphs that exhibit strong clusterability. Such graphs contain $k$ latent clusters, each characterized by a large inner conductance (at least $\varphi$) and a small outer conductance (at most $\varepsilon$). Our aim is to preprocess the graph to enable clustering membership queries, with the key requirement that both preprocessing and query answering should be performed in sublinear time, and the resulting partition should be consistent with a $k$-partition that is close to the ground-truth clustering. Previous oracles have relied on either a $\textrm{poly}(k)\log n$ gap between inner and outer conductances or exponential (in $k/\varepsilon$) preprocessing time.

An important component of the success of large AI models is double descent, in which networks avoid overfitting as they grow relative to the amount of training data, instead improving their performance on unseen data. Here we demonstrate double descent in a decentralized analog network of self-adjusting resistive elements. This system trains itself and performs tasks without a digital processor, offering potential gains in energy efficiency and speed -- but must endure component non-idealities. We find that standard training fails to yield double descent, but a modified protocol that accommodates this inherent imperfection succeeds. Our findings show that analog physical systems, if appropriately trained, can exhibit behaviors underlying the success of digital AI. Further, they suggest that biological systems might similarly benefit from over-parameterization.

The algorithm takes as input a nonnegative tensor, which may be sparse, non-square, and asymmetric, and outputs subsets of indices from each dimension (co-clusters).

Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions.

In this study, we design a reservoir computing (RC) network by exploiting short- and long-term memory dynamics in Au/Ti/MoS$_2$/Au memristive devices. The temporal dynamics is engineered by controlling the thickness of the Chemical Vapor Deposited (CVD) MoS$_2$ films. Devices with a monolayer (1L)-MoS$_2$ film exhibit volatile (short-term memory) switching dynamics. We also report non-volatile resistance switching with excellent uniformity and analog behavior in conductance tuning for the multilayer (ML) MoS$_2$ memristive devices. We correlate this performance with trap-assisted space-charge limited conduction (SCLC) mechanism, leading to a bulk-limited resistance switching behavior. Four-bit reservoir states are generated using volatile memristors. The readout layer is implemented with an array of nonvolatile synapses. This small RC network achieves 89.56\% precision in a spoken-digit recognition task and is also used to analyze a nonlinear time series equation.

Sparse and stochastic graphs create several "dangling

Sparse and stochastic graphs create several "dangling

Table C.2: Summary of ground-truth clusters used in the experiments