Long-term, repeated measurements of individual synaptic properties have revealed that synapses can undergo significant directed and spontaneous changes over time scales of minutes to weeks. These changes are presumably driven by a large number of activity-dependent and independent molecular processes, yet how these processes integrate to determine the totality of synaptic size remains unknown. Here we propose, as an alternative to detailed, mechanistic descriptions, a statistical approach to synaptic size dynamics. The basic premise of this approach is that the integrated outcome of the myriad of processes that drive synaptic size dynamics are effectively described as a combination of multiplicative and additive processes, both of which are stochastic and taken from distributions parametrically affected by physiological signals. We show that this seemingly simple model, known in probability theory as the Kesten process, can generate rich dynamics which are qualitatively similar to the dynamics of individual glutamatergic synapses recorded in long-term time-lapse experiments in ex-vivo cortical networks. Moreover, we show that this stochastic model, which is insensitive to many of its underlying details, quantitatively captures the distributions of synaptic sizes measured in these experiments, the long-term stability of such distributions and their scaling in response to pharmacological manipulations. Finally, we show that the average kinetics of new postsynaptic density formation measured in such experiments is also faithfully captured by the same model. The model thus provides a useful framework for characterizing synapse size dynamics at steady state, during initial formation of such steady states, and during their convergence to new steady states following perturbations. These findings show the strength of a simple low dimensional statistical model to quantitatively describe synapse size dynamics as the integrated result of many underlying complex processes.
The k-means problem is to compute a set of k centers (points) that minimizes the sum of squared distances to a given set of n points in a metric space. Arguably, the most common algorithm to solve it is k-means++ which is easy to implement and provides a provably small approximation error in time that is linear in n. We generalize k-means++ to support outliers in two sense (simultaneously): (i) nonmetric spaces, e.g., M-estimators, where the distance dist(p,x) between a point p and a center x is replaced by mindist(p,x),c for an appropriate constant c that may depend on the scale of the input. (ii) k-means clustering with m≥1 outliers, i.e., where the m farthest points from any given k centers are excluded from the total sum of distances. This is by using a simple reduction to the (k+m)-means clustering (with no outliers).
An ε-coreset for Least-Mean-Squares (LMS) of a matrix A ∈ R n×d is a small weighted subset of its rows that approximates the sum of squared distances from its rows to every affine k-dimensional subspace of R d , up to a factor of 1±ε. Such coresets are useful for hyper-parameter tuning and solving many least-mean-squares problems such as low-rank approximation (k-SVD), k-PCA, Lassso/Ridge/Linear regression and many more. Coresets are also useful for handling streaming, dynamic and distributed big data in parallel. With high probability, non-uniform sampling based on upper bounds on what is known as importance or sensitivity of each row in A yields a coreset. The size of the (sampled) coreset is then near-linear in the total sum of these sensitivity bounds.We provide algorithms that compute provably tight bounds for the sensitivity of each input row.It is based on two ingredients: (i) iterative algorithm that computes the exact sensitivity of each point up to arbitrary small precision for (non-affine) k-subspaces, and (ii) a general reduction of independent interest from computing sensitivity for the family of affine k-subspaces in R d to (non-affine) (k + 1)-subspaces in R d+1 .Experimental results on real-world datasets, including the English Wikipedia documentsterm matrix, show that our bounds provide significantly smaller and data-dependent coresets also in practice. Full open source is also provided.
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