similarity_graph {uwot}

Similarity Graph

Description

Create a graph (as a sparse symmetric weighted adjacency matrix) representing the similarities between items in a data set. No dimensionality reduction is carried out. By default, the similarities are calculated using the merged fuzzy simplicial set approach in the Uniform Manifold Approximation and Projection (UMAP) method (McInnes et al., 2018), but the approach from LargeVis (Tang et al., 2016) can also be used.

Usage

similarity_graph(
  X = NULL,
  n_neighbors = NULL,
  metric = "euclidean",
  scale = NULL,
  set_op_mix_ratio = 1,
  local_connectivity = 1,
  nn_method = NULL,
  n_trees = 50,
  search_k = 2 * n_neighbors * n_trees,
  perplexity = 50,
  method = "umap",
  y = NULL,
  target_n_neighbors = n_neighbors,
  target_metric = "euclidean",
  target_weight = 0.5,
  pca = NULL,
  pca_center = TRUE,
  ret_extra = c(),
  n_threads = NULL,
  grain_size = 1,
  kernel = "gauss",
  tmpdir = tempdir(),
  verbose = getOption("verbose", TRUE),
  pca_method = NULL,
  binary_edge_weights = FALSE,
  nn_args = list()
)

Arguments

X

Input data. Can be a data.frame, matrix, dist object or sparseMatrix. Matrix and data frames should contain one observation per row. Data frames will have any non-numeric columns removed, although factor columns will be used if explicitly included via metric (see the help for metric for details). A sparse matrix is interpreted as a distance matrix, and is assumed to be symmetric, so you can also pass in an explicitly upper or lower triangular sparse matrix to save storage. There must be at least n_neighbors non-zero distances for each row. Both implicit and explicit zero entries are ignored. Set zero distances you want to keep to an arbitrarily small non-zero value (e.g. 1e-10). X can also be NULL if pre-computed nearest neighbor data is passed to nn_method.

n_neighbors

The size of local neighborhood (in terms of number of neighboring sample points) used for manifold approximation. Larger values result in more global views of the manifold, while smaller values result in more local data being preserved. In general values should be in the range 2 to 100.

metric

Type of distance metric to use to find nearest neighbors. For nn_method = "annoy" this can be one of:

• "euclidean" (the default)

• "cosine"

• "manhattan"

• "hamming"

• "correlation" (a distance based on the Pearson correlation)

• "categorical" (see below)

For nn_method = "hnsw" this can be one of:

• "euclidean"

• "cosine"

• "correlation"

If rnndescent is installed and nn_method = "nndescent" is specified then many more metrics are avaiable, including:

• "braycurtis"

• "canberra"

• "chebyshev"

• "dice"

• "hamming"

• "hellinger"

• "jaccard"

• "jensenshannon"

• "kulsinski"

• "rogerstanimoto"

• "russellrao"

• "sokalmichener"

• "sokalsneath"

• "spearmanr"

• "symmetrickl"

• "tsss"

• "yule"

For more details see the package documentation of rnndescent. For nn_method = "fnn", the distance metric is always "euclidean".

If X is a data frame or matrix, then multiple metrics can be specified, by passing a list to this argument, where the name of each item in the list is one of the metric names above. The value of each list item should be a vector giving the names or integer ids of the columns to be included in a calculation, e.g. metric = list(euclidean = 1:4, manhattan = 5:10).

Each metric calculation results in a separate fuzzy simplicial set, which are intersected together to produce the final set. Metric names can be repeated. Because non-numeric columns are removed from the data frame, it is safer to use column names than integer ids.

Factor columns can also be used by specifying the metric name "categorical". Factor columns are treated different from numeric columns and although multiple factor columns can be specified in a vector, each factor column specified is processed individually. If you specify a non-factor column, it will be coerced to a factor.

For a given data block, you may override the pca and pca_center arguments for that block, by providing a list with one unnamed item containing the column names or ids, and then any of the pca or pca_center overrides as named items, e.g. metric = list(euclidean = 1:4, manhattan = list(5:10, pca_center = FALSE)). This exists to allow mixed binary and real-valued data to be included and to have PCA applied to both, but with centering applied only to the real-valued data (it is typical not to apply centering to binary data before PCA is applied).

scale

Scaling to apply to X if it is a data frame or matrix:

• "none" or FALSE or NULL No scaling.

• "Z" or "scale" or TRUE Scale each column to zero mean and variance 1.

• "maxabs" Center each column to mean 0, then divide each element by the maximum absolute value over the entire matrix.

• "range" Range scale the entire matrix, so the smallest element is 0 and the largest is 1.

• "colrange" Scale each column in the range (0,1).

For method "umap", the default is "none". For "largevis", the default is "maxabs".

set_op_mix_ratio

Interpolate between (fuzzy) union and intersection as the set operation used to combine local fuzzy simplicial sets to obtain a global fuzzy simplicial sets. Both fuzzy set operations use the product t-norm. The value of this parameter should be between 0.0 and 1.0; a value of 1.0 will use a pure fuzzy union, while 0.0 will use a pure fuzzy intersection. Ignored if method = "largevis"

local_connectivity

The local connectivity required – i.e. the number of nearest neighbors that should be assumed to be connected at a local level. The higher this value the more connected the manifold becomes locally. In practice this should be not more than the local intrinsic dimension of the manifold. Ignored if method = "largevis".

nn_method

Method for finding nearest neighbors. Options are:

• "fnn". Use exact nearest neighbors via the FNN package.

• "annoy" Use approximate nearest neighbors via the RcppAnnoy package.

• "hnsw" Use approximate nearest neighbors with the Hierarchical Navigable Small World (HNSW) method (Malkov and Yashunin, 2018) via the RcppHNSW package. RcppHNSW is not a dependency of this package: this option is only available if you have installed RcppHNSW yourself. Also, HNSW only supports the following arguments for metric and target_metric: "euclidean", "cosine" and "correlation".

• "nndescent" Use approximate nearest neighbors with the Nearest Neighbor Descent method (Dong et al., 2011) via the rnndescent package. rnndescent is not a dependency of this package: this option is only available if you have installed rnndescent yourself.

By default, if X has less than 4,096 vertices, the exact nearest neighbors are found. Otherwise, approximate nearest neighbors are used. You may also pass pre-calculated nearest neighbor data to this argument. It must be one of two formats, either a list consisting of two elements:

• "idx". A n_vertices x n_neighbors matrix containing the integer indexes of the nearest neighbors in X. Each vertex is considered to be its own nearest neighbor, i.e. idx[, 1] == 1:n_vertices.

• "dist". A n_vertices x n_neighbors matrix containing the distances of the nearest neighbors.

or a sparse distance matrix of type dgCMatrix, with dimensions n_vertices x n_vertices. Distances should be arranged by column, i.e. a non-zero entry in row j of the ith column indicates that the jth observation in X is a nearest neighbor of the ith observation with the distance given by the value of that element. The n_neighbors parameter is ignored when using precomputed nearest neighbor data. If using the sparse distance matrix input, each column can contain a different number of neighbors.

n_trees

Number of trees to build when constructing the nearest neighbor index. The more trees specified, the larger the index, but the better the results. With search_k, determines the accuracy of the Annoy nearest neighbor search. Only used if the nn_method is "annoy". Sensible values are between 10 to 100.

search_k

Number of nodes to search during the neighbor retrieval. The larger k, the more the accurate results, but the longer the search takes. With n_trees, determines the accuracy of the Annoy nearest neighbor search. Only used if the nn_method is "annoy".

perplexity

Used only if method = "largevis". Controls the size of the local neighborhood used for manifold approximation. Should be a value between 1 and one less than the number of items in X. If specified, you should not specify a value for n_neighbors unless you know what you are doing.

method

How to generate the similarities between items. One of:

• "umap" The UMAP method of McInnes et al. (2018).

• "largevis" The LargeVis method of Tang et al. (2016).

y

Optional target data to add supervised or semi-supervised weighting to the similarity graph . Can be a vector, matrix or data frame. Use the target_metric parameter to specify the metrics to use, using the same syntax as metric. Usually either a single numeric or factor column is used, but more complex formats are possible. The following types are allowed:

• Factor columns with the same length as X. NA is allowed for any observation with an unknown level, in which case UMAP operates as a form of semi-supervised learning. Each column is treated separately.

• Numeric data. NA is not allowed in this case. Use the parameter target_n_neighbors to set the number of neighbors used with y. If unset, n_neighbors is used. Unlike factors, numeric columns are grouped into one block unless target_metric specifies otherwise. For example, if you wish columns a and b to be treated separately, specify target_metric = list(euclidean = "a", euclidean = "b"). Otherwise, the data will be effectively treated as a matrix with two columns.

• Nearest neighbor data, consisting of a list of two matrices, idx and dist. These represent the precalculated nearest neighbor indices and distances, respectively. This is the same format as that expected for precalculated data in nn_method. This format assumes that the underlying data was a numeric vector. Any user-supplied value of the target_n_neighbors parameter is ignored in this case, because the the number of columns in the matrices is used for the value. Multiple nearest neighbor data using different metrics can be supplied by passing a list of these lists.

Unlike X, all factor columns included in y are automatically used. This parameter is ignored if method = "largevis".

target_n_neighbors

Number of nearest neighbors to use to construct the target simplicial set. Default value is n_neighbors. Applies only if y is non-NULL and numeric. This parameter is ignored if method = "largevis".

target_metric

The metric used to measure distance for y if using supervised dimension reduction. Used only if y is numeric. This parameter is ignored if method = "largevis".

target_weight

Weighting factor between data topology and target topology. A value of 0.0 weights entirely on data, a value of 1.0 weights entirely on target. The default of 0.5 balances the weighting equally between data and target. Only applies if y is non-NULL. This parameter is ignored if method = "largevis".

pca

If set to a positive integer value, reduce data to this number of columns using PCA. Doesn't applied if the distance metric is "hamming", or the dimensions of the data is larger than the number specified (i.e. number of rows and columns must be larger than the value of this parameter). If you have > 100 columns in a data frame or matrix, reducing the number of columns in this way may substantially increase the performance of the nearest neighbor search at the cost of a potential decrease in accuracy. In many t-SNE applications, a value of 50 is recommended, although there's no guarantee that this is appropriate for all settings.

pca_center

If TRUE, center the columns of X before carrying out PCA. For binary data, it's recommended to set this to FALSE.

ret_extra

A vector indicating what extra data to return. May contain any combination of the following strings:

• "nn" nearest neighbor data that can be used as input to nn_method to avoid the overhead of repeatedly calculating the nearest neighbors when manipulating unrelated parameters. See the "Value" section for the names of the list items. Note that the nearest neighbors could be sensitive to data scaling, so be wary of reusing nearest neighbor data if modifying the scale parameter.

• "sigma" the normalization value for each observation in the dataset when constructing the smoothed distances to each of its neighbors. This gives some sense of the local density of each observation in the high dimensional space: higher values of sigma indicate a higher dispersion or lower density.

n_threads

Number of threads to use. Default is half the number of concurrent threads supported by the system. For nearest neighbor search, only applies if nn_method = "annoy". If n_threads > 1, then the Annoy index will be temporarily written to disk in the location determined by tempfile.

grain_size

The minimum amount of work to do on each thread. If this value is set high enough, then less than n_threads will be used for processing, which might give a performance improvement if the overhead of thread management and context switching was outweighing the improvement due to concurrent processing. This should be left at default (1) and work will be spread evenly over all the threads specified.

kernel

Used only if method = "largevis". Type of kernel function to create input similiarties. Can be one of "gauss" (the default) or "knn". "gauss" uses the usual Gaussian weighted similarities. "knn" assigns equal similiarties. to every edge in the nearest neighbor graph, and zero otherwise, using perplexity nearest neighbors. The n_neighbors parameter is ignored in this case.

tmpdir

Temporary directory to store nearest neighbor indexes during nearest neighbor search. Default is tempdir. The index is only written to disk if n_threads > 1 and nn_method = "annoy"; otherwise, this parameter is ignored.

verbose

If TRUE, log details to the console.

pca_method

Method to carry out any PCA dimensionality reduction when the pca parameter is specified. Allowed values are:

• "irlba". Uses prcomp_irlba from the irlba package.

• "rsvd". Uses 5 iterations of svdr from the irlba package. This is likely to give much faster but potentially less accurate results than using "irlba". For the purposes of nearest neighbor calculation and coordinates initialization, any loss of accuracy doesn't seem to matter much.

• "bigstatsr". Uses big_randomSVD from the bigstatsr package. The SVD methods used in bigstatsr may be faster on systems without access to efficient linear algebra libraries (e.g. Windows). Note: bigstatsr is not a dependency of uwot: if you choose to use this package for PCA, you must install it yourself.

• "svd". Uses svd for the SVD. This is likely to be slow for all but the smallest datasets.

• "auto" (the default). Uses "irlba", unless more than 50 case "svd" is used.

binary_edge_weights

If TRUE then edge weights of the returned graph are binary (0/1) rather than reflecting the degree of similarity.

nn_args

A list containing additional arguments to pass to the nearest neighbor method. For nn_method = "annoy", you can specify "n_trees" and "search_k", and these will override the n_trees and search_k parameters. For nn_method = "hnsw", you may specify the following arguments:

• M The maximum number of neighbors to keep for each vertex. Reasonable values are 2 to 100. Higher values give better recall at the cost of more memory. Default value is 16.

• ef_construction A positive integer specifying the size of the dynamic list used during index construction. A higher value will provide better results at the cost of a longer time to build the index. Default is 200.

• ef A positive integer specifying the size of the dynamic list used during search. This cannot be smaller than n_neighbors and cannot be higher than the number of items in the index. Default is 10.

For nn_method = "nndescent", you may specify the following arguments:

• n_trees The number of trees to use in a random projection forest to initialize the search. A larger number will give more accurate results at the cost of a longer computation time. The default of NULL means that the number is chosen based on the number of observations in X.

• max_candidates The number of potential neighbors to explore per iteration. By default, this is set to n_neighbors or 60, whichever is smaller. A larger number will give more accurate results at the cost of a longer computation time.

• n_iters The number of iterations to run the search. A larger number will give more accurate results at the cost of a longer computation time. By default, this will be chosen based on the number of observations in X. You may also need to modify the convergence criterion delta.

• delta The minimum relative change in the neighbor graph allowed before early stopping. Should be a value between 0 and 1. The smaller the value, the smaller the amount of progress between iterations is allowed. Default value of 0.001 means that at least 0.1 neighbor graph must be updated at each iteration.

• init How to initialize the nearest neighbor descent. By default this is set to "tree" and uses a random project forest. If you set this to "rand", then a random selection is used. Usually this is less accurate than using RP trees, but for high-dimensional cases, there may be little difference in the quality of the initialization and random initialization will be a lot faster. If you set this to "rand", then the n_trees parameter is ignored.

• pruning_degree_multiplier The maximum number of edges per node to retain in the search graph, relative to n_neighbors. A larger value will give more accurate results at the cost of a longer computation time. Default is 1.5. This parameter only affects neighbor search when transforming new data with umap_transform.

• epsilon Controls the degree of the back-tracking when traversing the search graph. Setting this to 0.0 will do a greedy search with no back-tracking. A larger value will give more accurate results at the cost of a longer computation time. Default is 0.1. This parameter only affects neighbor search when transforming new data with umap_transform.

• max_search_fraction Specifies the maximum fraction of the search graph to traverse. By default, this is set to 1.0, so the entire graph (i.e. all items in X) may be visited. You may want to set this to a smaller value if you have a very large dataset (in conjunction with epsilon) to avoid an inefficient exhaustive search of the data in X. This parameter only affects neighbor search when transforming new data with umap_transform.

Details

This is equivalent to running umap with the ret_extra = c("fgraph") parameter, but without the overhead of calculating (or returning) the optimized low-dimensional coordinates.

Value

A sparse symmetrized matrix of the similarities between the items in X or if nn_method contains pre-computed nearest neighbor data, the items in nn_method. Because of the symmetrization, there may be more non-zero items in each column than the specified value of n_neighbors (or pre-computed neighbors in nn_method). If ret_extra is specified then the return value will be a list containing:

• similarity_graph the similarity graph as a sparse matrix as described above.

• nn (if ret_extra contained "nn") the nearest neighbor data as a list called nn. This contains one list for each metric calculated, itself containing a matrix idx with the integer ids of the neighbors; and a matrix dist with the distances. The nn list (or a sub-list) can be used as input to the nn_method parameter.

• sigma (if ret_extra contains "sigma"), a vector of calibrated parameters, one for each item in the input data, reflecting the local data density for that item. The exact definition of the values depends on the choice of the method parameter.

• rho (if ret_extra contains "sigma"), a vector containing the largest distance to the locally connected neighbors of each item in the input data. This will exist only if method = "umap".

• localr (if ret_extra contains "localr") a vector of the estimated local radii, the sum of "sigma" and "rho". This will exist only if method = "umap".

References

Dong, W., Moses, C., & Li, K. (2011, March). Efficient k-nearest neighbor graph construction for generic similarity measures. In Proceedings of the 20th international conference on World Wide Web (pp. 577-586). ACM. doi:10.1145/1963405.1963487.

Malkov, Y. A., & Yashunin, D. A. (2018). Efficient and robust approximate nearest neighbor search using hierarchical navigable small world graphs. IEEE transactions on pattern analysis and machine intelligence, 42(4), 824-836.

McInnes, L., Healy, J., & Melville, J. (2018). UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction arXiv preprint arXiv:1802.03426. https://arxiv.org/abs/1802.03426

Tang, J., Liu, J., Zhang, M., & Mei, Q. (2016, April). Visualizing large-scale and high-dimensional data. In Proceedings of the 25th International Conference on World Wide Web (pp. 287-297). International World Wide Web Conferences Steering Committee. https://arxiv.org/abs/1602.00370

Examples

iris30 <- iris[c(1:10, 51:60, 101:110), ]

# return a 30 x 30 sparse matrix with similarity data based on 10 nearest
# neighbors per item
iris30_sim_graph <- similarity_graph(iris30, n_neighbors = 10)

# Default is to use the UMAP method of calculating similarities, but LargeVis
# is also available: for that method, use perplexity instead of n_neighbors
# to control neighborhood size. Use ret_extra = "nn" to return nearest
# neighbor data as well as the similarity graph. Return value is a list
# containing similarity_graph' and 'nn' items.
iris30_lv_graph <- similarity_graph(iris30,
  perplexity = 10,
  method = "largevis", ret_extra = "nn"
)
# If you have the neighbor information you don't need the original data
iris30_lv_graph_nn <- similarity_graph(
  nn_method = iris30_lv_graph$nn,
  perplexity = 10, method = "largevis"
)
all(iris30_lv_graph_nn == iris30_lv_graph$similarity_graph)