Sketch-based Randomized Algorithms for Dynamic Graph Regression

A well-known problem in data science and machine learning is {\em linear regression}, which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression problem require at least a linear time (in terms of $n$: the size of the graph). However, this time complexity might be intractable in practice. In the current paper, we utilize {\em subsampled randomized Hadamard transform} and \textsf{CountSketch} to propose the first randomized algorithms. Suppose that we are given an $n\times m$ matrix embedding $M$ of the graph, where $m \ll n$. Let $r$ be the number of samples required for a guaranteed approximation error, which is a sublinear function of $n$. Our first algorithm reduces time complexity of pre-processing to $O(n(m + 1) + 2n(m + 1) \log_2(r + 1) + rm^2)$. Then after an edge insertion or an edge deletion, it updates the approximate solution in $O(rm)$ time. Our second algorithm reduces time complexity of pre-processing to $O \left( nnz(M) + m^3 \epsilon^{-2} \log^7(m/\epsilon) \right)$, where $nnz(M)$ is the number of nonzero elements of $M$. Then after an edge insertion or an edge deletion or a node insertion or a node deletion, it updates the approximate solution in $O(qm)$ time, with $q=O\left(\frac{m^2}{\epsilon^2} \log^6(m/\epsilon) \right)$. Finally, we show that under some assumptions, if $\ln n < \epsilon^{-1}$ our first algorithm outperforms our second algorithm and if $\ln n \geq \epsilon^{-1}$ our second algorithm outperforms our first algorithm

Exact algorithms for L1L^1-TV regularization of real-valued or circle-valued signals

We consider $L^1$-TV regularization of univariate signals with values on the real line or on the unit circle. While the real data space leads to a convex optimization problem, the problem is non-convex for circle-valued data. In this paper, we derive exact algorithms for both data spaces. A key ingredient is the reduction of the infinite search spaces to a finite set of configurations, which can be scanned by the Viterbi algorithm. To reduce the computational complexity of the involved tabulations, we extend the technique of distance transforms to non-uniform grids and to the circular data space. In total, the proposed algorithms have complexity $\mathscr{O}(KN)$ where $N$ is the length of the signal and $K$ is the number of different values in the data set. In particular, the complexity is $\mathscr{O}(N)$ for quantized data. It is the first exact algorithm for TV regularization with circle-valued data, and it is competitive with the state-of-the-art methods for scalar data, assuming that the latter are quantized