projectKNNs {pagoda2}

Project a distance matrix into a lower-dimensional space. (from elbamos/largeVis)

Description

Takes as input a sparse matrix of the edge weights connecting each node to its nearest neighbors, and outputs a matrix of coordinates embedding the inputs in a lower-dimensional space.

Usage

projectKNNs(
  wij,
  dim = 2,
  sgd_batches = NULL,
  M = 5,
  gamma = 7,
  alpha = 1,
  rho = 1,
  coords = NULL,
  useDegree = FALSE,
  momentum = NULL,
  seed = NULL,
  threads = NULL,
  verbose = getOption("verbose", TRUE)
)

Arguments

wij

A symmetric sparse matrix of edge weights, in C-compressed format, as created with the Matrix package.

dim

numeric The number of dimensions for the projection space (default=2)

sgd_batches

numeric The number of edges to process during SGD (default=NULL). Defaults to a value set based on the size of the dataset. If the parameter given is between 0 and 1, the default value will be multiplied by the parameter.

M

numeric (largeVis) The number of negative edges to sample for each positive edge (default=5).

gamma

numeric (largeVis) The strength of the force pushing non-neighbor nodes apart (default=7).

alpha

numeric (largeVis) The hyperparameter in the distance function (default=1). The default distance function, 1 / (1 + \alpha \dot ||y_i - y_j||^2). The function relates the distance between points in the low-dimensional projection to the likelihood that the two points are nearest neighbors. Increasing \alpha tends to push nodes and their neighbors closer together; decreasing \alpha produces a broader distribution. Setting \alpha to zero enables the alternative distance function. \alpha below zero is meaningless.

rho

(largeVis) numeric Initial learning rate (default=1)

coords

An initialized coordinate matrix (default=NULL)

useDegree

boolean Whether to use vertex degree to determine weights in negative sampling (if TRUE) or the sum of the vertex's edges (if FALSE) (default=FALSE)

momentum

If not NULL, SGD with momentum is used, with this multiplier, which must be between 0 and 1 (default=NULL). Note that momentum can drastically speed-up training time, at the cost of additional memory consumed.

seed

numeric Random seed to be passed to the C++ functions (default=NULL). Sampled from hardware entropy pool if NULL (the default). Note that if the seed is not NULL (the default), the maximum number of threads will be set to 1 in phases of the algorithm that would otherwise be non-deterministic.

threads

numeric The maximum number of threads to spawn (default=NULL). Determined automatically if NULL (the default).

verbose

boolean Verbosity (default=getOption("verbose", TRUE))

Details

The algorithm attempts to estimate a dim-dimensional embedding using stochastic gradient descent and negative sampling.

The objective function is:

O = \sum_{(i,j)\in E} w_{ij} (\log f(||p(e_{ij} = 1||) + \sum_{k=1}^{M} E_{jk~P_{n}(j)} \gamma \log(1 - f(||p(e_{ij_k} - 1||)))

where f() is a probabilistic function relating the distance between two points in the low-dimensional projection space, and the probability that they are nearest neighbors.

The default probabilistic function is 1 / (1 + \alpha \dot ||x||^2). If \alpha is set to zero, an alternative probabilistic function, 1 / (1 + \exp(x^2)) will be used instead.

Note that the input matrix should be symmetric. If any columns in the matrix are empty, the function will fail.

Value

A dense [N,D] matrix of the coordinates projecting the w_ij matrix into the lower-dimensional space.

Note

If specified, seed is passed to the C++ and used to initialize the random number generator. This will not, however, be sufficient to ensure reproducible results, because the initial coordinate matrix is generated using the R random number generator. To ensure reproducibility, call set.seed before calling this function, or pass it a pre-allocated coordinate matrix.

The original paper called for weights in negative sampling to be calculated according to the degree of each vertex, the number of edges connecting to the vertex. The reference implementation, however, uses the sum of the weights of the edges to each vertex. In experiments, the difference was imperceptible with small (MNIST-size) datasets, but the results seems aesthetically preferrable using degree. The default is to use the edge weights, consistent with the reference implementation.