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Dimensionality Reduction with a LargeVis-like method

Source: R/uwot.R
lvish.Rd

Carry out dimensionality reduction of a dataset using a method similar to LargeVis (Tang et al., 2016).

Usage

lvish(
  X,
  perplexity = 50,
  n_neighbors = perplexity * 3,
  n_components = 2,
  metric = "euclidean",
  n_epochs = -1,
  learning_rate = 1,
  scale = "maxabs",
  init = "lvrandom",
  init_sdev = NULL,
  repulsion_strength = 7,
  negative_sample_rate = 5,
  nn_method = NULL,
  n_trees = 50,
  search_k = 2 * n_neighbors * n_trees,
  n_threads = NULL,
  n_sgd_threads = 0,
  grain_size = 1,
  kernel = "gauss",
  pca = NULL,
  pca_center = TRUE,
  pcg_rand = TRUE,
  fast_sgd = FALSE,
  ret_nn = FALSE,
  ret_extra = c(),
  tmpdir = tempdir(),
  verbose = getOption("verbose", TRUE),
  batch = FALSE,
  opt_args = NULL,
  epoch_callback = NULL,
  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, and init is not "spca" or "pca".

perplexity

Controls the size of the local neighborhood used for manifold approximation. This is the analogous to n_neighbors in umap. Change this, rather than n_neighbors.

n_neighbors

The number of neighbors to use when calculating the perplexity. Usually set to three times the value of the perplexity. Must be at least as large as perplexity.

n_components

The dimension of the space to embed into. This defaults to 2 to provide easy visualization, but can reasonably be set to any integer value 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).

n_epochs

Number of epochs to use during the optimization of the embedded coordinates. The default is calculate the number of epochs dynamically based on dataset size, to give the same number of edge samples as the LargeVis defaults. This is usually substantially larger than the UMAP defaults. If n_epochs = 0, then coordinates determined by "init" will be returned.

learning_rate

Initial learning rate used in optimization of the coordinates.

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 lvish, the default is "maxabs", for consistency with LargeVis.

init

Type of initialization for the coordinates. Options are:

  • "spectral" Spectral embedding using the normalized Laplacian of the fuzzy 1-skeleton, with Gaussian noise added.

  • "normlaplacian". Spectral embedding using the normalized Laplacian of the fuzzy 1-skeleton, without noise.

  • "random". Coordinates assigned using a uniform random distribution between -10 and 10.

  • "lvrandom". Coordinates assigned using a Gaussian distribution with standard deviation 1e-4, as used in LargeVis (Tang et al., 2016) and t-SNE.

  • "laplacian". Spectral embedding using the Laplacian Eigenmap (Belkin and Niyogi, 2002).

  • "pca". The first two principal components from PCA of X if X is a data frame, and from a 2-dimensional classical MDS if X is of class "dist".

  • "spca". Like "pca", but each dimension is then scaled so the standard deviation is 1e-4, to give a distribution similar to that used in t-SNE and LargeVis. This is an alias for init = "pca", init_sdev = 1e-4.

  • "agspectral" An "approximate global" modification of "spectral" which all edges in the graph to a value of 1, and then sets a random number of edges (negative_sample_rate edges per vertex) to 0.1, to approximate the effect of non-local affinities.

  • A matrix of initial coordinates.

For spectral initializations, ("spectral", "normlaplacian", "laplacian", "agspectral"), if more than one connected component is identified, no spectral initialization is attempted. Instead a PCA-based initialization is attempted. If verbose = TRUE the number of connected components are logged to the console. The existence of multiple connected components implies that a global view of the data cannot be attained with this initialization. Increasing the value of n_neighbors may help.

init_sdev

If non-NULL, scales each dimension of the initialized coordinates (including any user-supplied matrix) to this standard deviation. By default no scaling is carried out, except when init = "spca", in which case the value is 0.0001. Scaling the input may help if the unscaled versions result in initial coordinates with large inter-point distances or outliers. This usually results in small gradients during optimization and very little progress being made to the layout. Shrinking the initial embedding by rescaling can help under these circumstances. Scaling the result of init = "pca" is usually recommended and init = "spca" as an alias for init = "pca", init_sdev = 1e-4 but for the spectral initializations the scaled versions usually aren't necessary unless you are using a large value of n_neighbors (e.g. n_neighbors = 150 or higher). For compatibility with recent versions of the Python UMAP package, if you are using init = "spectral", then you should also set init_sdev = "range", which will range scale each of the columns containing the initial data between 0-10. This is not set by default to maintain backwards compatibility with previous versions of uwot.

repulsion_strength

Weighting applied to negative samples in low dimensional embedding optimization. Values higher than one will result in greater weight being given to negative samples.

negative_sample_rate

The number of negative edge/1-simplex samples to use per positive edge/1-simplex sample in optimizing the low dimensional embedding.

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: "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 precalculated nearest neighbor data to this argument. It must be 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.

Multiple nearest neighbor data (e.g. from two different precomputed metrics) can be passed by passing a list containing the nearest neighbor data lists as items. The n_neighbors parameter is ignored when using precomputed nearest neighbor data.

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".

n_threads

Number of threads to use (except during stochastic gradient descent). 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.

n_sgd_threads

Number of threads to use during stochastic gradient descent. If set to > 1, then be aware that if batch = FALSE, results will not be reproducible, even if set.seed is called with a fixed seed before running. Set to "auto" to use the same value as n_threads.

grain_size

The minimum amount of work to do on each thread. If this value is set high enough, then less than n_threads or n_sgd_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

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

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.

pcg_rand

If TRUE, use the PCG random number generator (O'Neill, 2014) during optimization. Otherwise, use the faster (but probably less statistically good) Tausworthe "taus88" generator. The default is TRUE.

fast_sgd

If TRUE, then the following combination of parameters is set: pcg_rand = TRUE and n_sgd_threads = "auto". The default is FALSE. Setting this to TRUE will speed up the stochastic optimization phase, but give a potentially less accurate embedding, and which will not be exactly reproducible even with a fixed seed. For visualization, fast_sgd = TRUE will give perfectly good results. For more generic dimensionality reduction, it's safer to leave fast_sgd = FALSE. If fast_sgd = TRUE, then user-supplied values of pcg_rand and n_sgd_threads, are ignored.

ret_nn

If TRUE, then in addition to the embedding, also return 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 (e.g. min_dist, n_epochs, init). See the "Value" section for the names of the list items. If FALSE, just return the coordinates. Note that the nearest neighbors could be sensitive to data scaling, so be wary of reusing nearest neighbor data if modifying the scale parameter.

ret_extra

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

  • "nn" same as setting ret_nn = TRUE.

  • "P" the high dimensional probability matrix. The graph is returned as a sparse symmetric N x N matrix of class dgCMatrix-class, where a non-zero entry (i, j) gives the input probability (or similarity or affinity) of the edge connecting vertex i and vertex j. Note that the graph is further sparsified by removing edges with sufficiently low membership strength that they would not be sampled by the probabilistic edge sampling employed for optimization and therefore the number of non-zero elements in the matrix is dependent on n_epochs. If you are only interested in the fuzzy input graph (e.g. for clustering), setting n_epochs = 0 will avoid any further sparsifying. Be aware that setting binary_edge_weights = TRUE will affect this graph (all non-zero edge weights will be 1).

  • sigma a vector of the bandwidths used to calibrate the input Gaussians to reproduce the target "perplexity".

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.

batch

If TRUE, then embedding coordinates are updated at the end of each epoch rather than during the epoch. In batch mode, results are reproducible with a fixed random seed even with n_sgd_threads > 1, at the cost of a slightly higher memory use. You may also have to modify learning_rate and increase n_epochs, so whether this provides a speed increase over the single-threaded optimization is likely to be dataset and hardware-dependent.

opt_args

A list of optimizer parameters, used when batch = TRUE. The default optimization method used is Adam (Kingma and Ba, 2014).

  • method The optimization method to use. Either "adam" or "sgd" (stochastic gradient descent). Default: "adam".

  • beta1 (Adam only). The weighting parameter for the exponential moving average of the first moment estimator. Effectively the momentum parameter. Should be a floating point value between 0 and 1. Higher values can smooth oscillatory updates in poorly-conditioned situations and may allow for a larger learning_rate to be specified, but too high can cause divergence. Default: 0.5.

  • beta2 (Adam only). The weighting parameter for the exponential moving average of the uncentered second moment estimator. Should be a floating point value between 0 and 1. Controls the degree of adaptivity in the step-size. Higher values put more weight on previous time steps. Default: 0.9.

  • eps (Adam only). Intended to be a small value to prevent division by zero, but in practice can also affect convergence due to its interaction with beta2. Higher values reduce the effect of the step-size adaptivity and bring the behavior closer to stochastic gradient descent with momentum. Typical values are between 1e-8 and 1e-3. Default: 1e-7.

  • alpha The initial learning rate. Default: the value of the learning_rate parameter.

epoch_callback

A function which will be invoked at the end of every epoch. Its signature should be: (epoch, n_epochs, coords), where:

  • epoch The current epoch number (between 1 and n_epochs).

  • n_epochs Number of epochs to use during the optimization of the embedded coordinates.

  • coords The embedded coordinates as of the end of the current epoch, as a matrix with dimensions (N, n_components).

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 in the input graph are treated as binary (0/1) rather than real valued. This affects the sampling frequency of neighbors and is the strategy used by the PaCMAP method (Wang and co-workers, 2020). Practical (Böhm and co-workers, 2020) and theoretical (Damrich and Hamprecht, 2021) work suggests this has little effect on UMAP's performance.

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.

Value

A matrix of optimized coordinates, or:

  • if ret_nn = TRUE (or ret_extra contains "nn"), returns 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.

  • if ret_extra contains "P", returns the high dimensional probability matrix as a sparse matrix called P, of type dgCMatrix-class.

  • if ret_extra contains "sigma", returns a vector of the high dimensional gaussian bandwidths for each point, and "dint" a vector of estimates of the intrinsic dimensionality at each point, based on the method given by Lee and co-workers (2015).

The returned list contains the combined data from any combination of specifying ret_nn and ret_extra.

Details

lvish differs from the official LargeVis implementation in the following:

  • Only the nearest-neighbor index search phase is multi-threaded.

  • Matrix input data is not normalized.

  • The n_trees parameter cannot be dynamically chosen based on data set size.

  • Nearest neighbor results are not refined via the neighbor-of-my-neighbor method. The search_k parameter is twice as large than default to compensate.

  • Gradient values are clipped to 4.0 rather than 5.0.

  • Negative edges are generated by uniform sampling of vertexes rather than their degree ^ 0.75.

  • The default number of samples is much reduced. The default number of epochs, n_epochs, is set to 5000, much larger than for umap, but may need to be increased further depending on your dataset. Using init = "spectral" can help.

References

Belkin, M., & Niyogi, P. (2002). Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in neural information processing systems (pp. 585-591). http://papers.nips.cc/paper/1961-laplacian-eigenmaps-and-spectral-techniques-for-embedding-and-clustering.pdf

Böhm, J. N., Berens, P., & Kobak, D. (2020). A unifying perspective on neighbor embeddings along the attraction-repulsion spectrum. arXiv preprint arXiv:2007.08902. https://arxiv.org/abs/2007.08902

Damrich, S., & Hamprecht, F. A. (2021). On UMAP's true loss function. Advances in Neural Information Processing Systems, 34. https://proceedings.neurips.cc/paper/2021/hash/2de5d16682c3c35007e4e92982f1a2ba-Abstract.html

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 .

Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. https://arxiv.org/abs/1412.6980

Lee, J. A., Peluffo-Ordóñez, D. H., & Verleysen, M. (2015). Multi-scale similarities in stochastic neighbour embedding: Reducing dimensionality while preserving both local and global structure. Neurocomputing, 169, 246-261.

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

O’Neill, M. E. (2014). PCG: A family of simple fast space-efficient statistically good algorithms for random number generation (Report No. HMC-CS-2014-0905). Harvey Mudd College.

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

Van der Maaten, L., & Hinton, G. (2008). Visualizing data using t-SNE. Journal of Machine Learning Research, 9 (2579-2605). https://www.jmlr.org/papers/v9/vandermaaten08a.html

Wang, Y., Huang, H., Rudin, C., & Shaposhnik, Y. (2021). Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMap, and PaCMAP for Data Visualization. Journal of Machine Learning Research, 22(201), 1-73. https://www.jmlr.org/papers/v22/20-1061.html

Examples

# Default number of epochs is much larger than for UMAP, assumes random
# initialization. Use perplexity rather than n_neighbors to control the size
# of the local neighborhood 20 epochs may be too small for a random
# initialization
iris_lvish <- lvish(iris,
  perplexity = 50, learning_rate = 0.5,
  init = "random", n_epochs = 20
)