We study a new model of computation, called best-order stream, for graph problems. Roughly, it is a proof system where a space-limited verifier has to verify a proof sequentially (i.e., it reads the proof as a stream). Moreover, the proof itself is just a specific ordering of the input data.

This model is closely related to many models of computation in other areas such as data streams, communication complexity, and proof checking, and could be used in applications such as cloud computing.

In this paper we focus on graph problems where the input is a sequence of edges. We show that even under this model, checking some basic graph properties deterministically requires linear space in the number of nodes. We also show that, in contrast with this, randomized verifiers are powerful enough to check many graph properties in polylogarithmic space.

## Another publication from the same category: Machine Learning and Data Science

### Regularization of the Kernel Matrix via Covariance Matrix Shrinkage Estimation

The kernel trick concept, formulated as an inner product in a feature space, facilitates powerful extensions to many well-known algorithms. While the kernel matrix involves inner products in the feature space, the sample covariance matrix of the data requires outer products. Therefore, their spectral properties are tightly connected. This allows us to examine the kernel matrix through the sample covariance matrix in the feature space and vice versa. The use of kernels often involves a large number of features, compared to the number of observations. In this scenario, the sample covariance matrix is not well-conditioned nor is it necessarily invertible, mandating a solution to the problem of estimating high-dimensional covariance matrices under small sample size conditions. We tackle this problem through the use of a shrinkage estimator that offers a compromise between the sample covariance matrix and a well-conditioned matrix (also known as the "target") with the aim of minimizing the mean-squared error (MSE). We propose a distribution-free kernel matrix regularization approach that is tuned directly from the kernel matrix, avoiding the need to address the feature space explicitly. Numerical simulations demonstrate that the proposed regularization is effective in classification tasks.

**Keywords**

**Categories**