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z_b9bO+!_;3WCS}HR1ax{{9)Ypl7TjJnwaweS@S93gC~xqK`X2eeolXiJ&q^&9u4tn zTH%L%FYaZ}wc(5T2_1BzR!-HXqkK5~0Kt4c_V-+et{NXhF z{pmOjT^oOg1D$Fo*^-b{GTDUZ{t?4d;v*|w!t#txvy(okt@KM)|372hwe!Sj(_G;F zb?h3#dT_km*EwsxFv^SV8vNj6t7tjc)cfBpDt_|-E5LE{A=;A-rlngy?*H}}Bm3B1 zn7!VNKfWGxw4%}$@3H6Tw~`h1+5>OL3yF-&&zB$>{om7R4vdD`xGUAt0u zwKH>4nqanwm3#Iw!P0UlVuZ)_7FLMkrY4mGItg74a`zg(3$?sMnrMOB(hq&jL>tW) zCtFe<&ifzevhJikpnWR$Q&(CzH)pqY!UnQ&V*PcMrS zPI7D_dfKIyC}|su@;zT5{YqMDGkjFjJ?_-(h- z^eOPchz8&ufcz*w|NN&50!HN2xa{1H$H>U&tT_0;F*5#FhS&d{Neuy@qe+f1>KNp4 zZpcR`+aXHDQiH98)Cp7f(8wU1L44Jhw2+&gKBS85w+SNE9>x@OOF{q_LFPj>@E1C{WIo6!su67lSj-La*r zbv&~ppD3kp6B+HooxJumj=kQO*f|T%XY7koN&&@d4@*Xg2p1#=gCRm;Wnp z5Y~&C;JN>9;&i=BbMU3dcy#cc8Oy~DgA zcSYpGSFDzZ3nn%<(O0ty|6S^DP+4P`=YFz2Q#@TfwRAw4n9{aBa!shj+~%!fprq8J vi3GXCGbGtWrWhy?5McAY)%pLJ{3zGh %\VignetteIndexEntry{uwot} %\VignetteEngine{knitr::rmarkdown} @@ -18,86 +21,250 @@ knitr::opts_chunk$set( library(uwot) ``` -```R -# See function man page for help -?umap - -# Non-numeric columns are ignored, so in a lot of cases you can pass a data -# frame directly to umap -iris_umap <- umap(iris, n_neighbors = 50, learning_rate = 0.5, init = "random") - -# Load mnist from somewhere, e.g. -# devtools::install_github("jlmelville/snedata") -# mnist <- snedata::download_mnist() -mnist_umap <- umap(mnist, n_neighbors = 15, min_dist = 0.001, verbose = TRUE) - -# For high dimensional datasets (> 100-1000 columns) using PCA to reduce -# dimensionality is highly recommended to avoid the nearest neighbor search -# taking a long time. Keeping only 50 dimensions can speed up calculations -# without affecting the visualization much -mnist_umap <- umap(mnist, pca = 50) - -# Use a specific number of threads -mnist_umap <- umap(mnist, n_neighbors = 15, min_dist = 0.001, verbose = TRUE, n_threads = 8) - -# Use a different metric -mnist_umap_cosine <- umap(mnist, n_neighbors = 15, metric = "cosine", min_dist = 0.001, verbose = TRUE, n_threads = 8) - -# If you are only interested in visualization, `fast_sgd = TRUE` gives a much faster optimization -mnist_umap_fast_sgd <- umap(mnist, n_neighbors = 15, metric = "cosine", min_dist = 0.001, verbose = TRUE, fast_sgd = TRUE) - -# Supervised dimension reduction -mnist_umap_s <- umap(mnist, n_neighbors = 15, min_dist = 0.001, verbose = TRUE, n_threads = 8, - y = mnist$Label, target_weight = 0.5) - -# Add new points to an existing embedding -mnist_train <- head(mnist, 60000) -mnist_test <- tail(mnist, 10000) - -# You must set ret_model = TRUE to return extra data we need -# coordinates are in mnist_train_umap$embedding -mnist_train_umap <- umap(mnist_train, verbose = TRUE, ret_model = TRUE) -mnist_test_umap <- umap_transform(mnist_test, mnist_train_umap, verbose = TRUE) - -# Save the nearest neighbor data -mnist_nn <- umap(mnist, ret_nn = TRUE) -# coordinates are now in mnist_nn$embedding - -# Re-use the nearest neighor data and save a lot of time -mnist_nn_spca <- umap(mnist, nn_method = mnist_nn$nn, init = "spca") - -# No problem to have ret_nn = TRUE and ret_model = TRUE at the same time -# Or just use the ret_extra parameter: -mnist_nn_and_model <- umap(mnist, ret_extra = c("model", "nn")) - -# You can also get to the input fuzzy graph as a sparse matrix via "fgraph" -mnist_with_fgraph <- umap(mnist, ret_extra = c("fgraph")) -# equivalent for lvish is to use "P" (input probability matrix): -mnist_with_P <- lvish(mnist, ret_extra = c("P")) - -# Calculate Petal and Sepal neighbors separately (uses intersection of the resulting sets): -iris_umap <- umap(iris, metric = list("euclidean" = c("Sepal.Length", "Sepal.Width"), - "euclidean" = c("Petal.Length", "Petal.Width"))) -# Can also use individual factor columns -iris_umap <- umap(iris, metric = list("euclidean" = c("Sepal.Length", "Sepal.Width"), - "euclidean" = c("Petal.Length", "Petal.Width"), - "categorical" = "Species")) - -# Batch mode allows for multiple threads while being reproducible -# (at the cost of needing more epochs for larger datasets) +`uwot` is the a package for implementing the UMAP dimensionality reduction +method. For more information on UMAP, see the +[original paper](https://arxiv.org/abs/1802.03426) and the +[Python package](https://github.com/lmcinnes/umap). + +We'll use the `iris` dataset in these examples. It's not the ideal dataset +because it's not terribly large nor high-dimensional (with only 4 numeric +columns), but you'll get the general idea. + +The default output dimensionality of UMAP is into two-dimensions, so it's +amenable for visualization, but you can set a larger value with `n_components`. +In this vignette we'll stick with two dimensions. We will need a function to +make plotting easier: + +```{r plot function} +kabsch <- function(pm, qm) { + pm_dims <- dim(pm) + if (!all(dim(qm) == pm_dims)) { + stop(call. = TRUE, "Point sets must have the same dimensions") + } + # The rotation matrix will have (ncol - 1) leading ones in the diagonal + diag_ones <- rep(1, pm_dims[2] - 1) + + # center the points + pm <- scale(pm, center = TRUE, scale = FALSE) + qm <- scale(qm, center = TRUE, scale = FALSE) + + am <- crossprod(pm, qm) + + svd_res <- svd(am) + # use the sign of the determinant to ensure a right-hand coordinate system + d <- determinant(tcrossprod(svd_res$v, svd_res$u))$sign + dm <- diag(c(diag_ones, d)) + + # rotation matrix + um <- svd_res$v %*% tcrossprod(dm, svd_res$u) + + # Rotate and then translate to the original centroid location of qm + sweep(t(tcrossprod(um, pm)), 2, -attr(qm, "scaled:center")) +} +iris_pca2 <- prcomp(iris[, 1:4])$x[, 1:2] +plot_umap <- function(coords, col = iris$Species, pca = iris_pca2) { + plot(kabsch(coords, pca), col = col, xlab = "", ylab = "") +} +``` + +Most of this code is just the +[kabsch algorithm](https://en.wikipedia.org/wiki/Kabsch_algorithm) +to align two point sets, which I am going to use to align the results of UMAP +over the first two principal components. This is to keep the relative +orientation of the output the same across different plots which makes it a bit +easier to see the differences between them. UMAP is a stochastic algorithm +so the output will be different each time you run it and small changes to the +parameters can affect the *absolute* values of the coordinates, although the +interpoint differences are usually similar. There's no need to go to such +trouble in most circumstances: the output of `umap` is a perfectly useful +2D matrix of coordinates you can pass into a plotting function with no further +processing required. + +## Basic UMAP + +The defaults of the `umap` function should work for most datasets. No scaling +of the input data is done, but non-numeric columns are ignored: + +```{r basic UMAP} +set.seed(42) +iris_umap <- umap(iris) +plot_umap(iris_umap) +``` + +### Parameters + +`uwot` has accumulated many parameters over time, but most of the time there +are only a handful you need worry about. The most important ones are: + +#### `min_dist` + +This is a mainly aesthetic parameter, which defines how close points can get in +the output space. A smaller value tends to make any clusters in the output more +compact. You should experiment with values between 0 and 1, although don't +choose exactly zero. The default is 0.01, which seems like it's a bit small for +`iris`. Let's crank up `min_dist` to `0.3`: + +```{r min_dist 0.5} +set.seed(42) +iris_umap_md05 <- umap(iris, min_dist = 0.3) +plot_umap(iris_umap_md05) +``` + +This has made the clusters bigger and closer together, so we'll use +`min_dist = 0.3` for the other examples with `iris`. + +#### `n_neighbors` + +This defines the number of items in the dataset that define the neighborhood +around each point. Set it too low and you will get a more fragmented layout. +Set it too high and you will get something that will miss a lot of local +structure. + +Here's a result with 5 neighbors: + +```{r 5 neighbors} +set.seed(42) +iris_umap_nbrs5 <- umap(iris, n_neighbors = 5, min_dist = 0.3) +plot_umap(iris_umap_nbrs5) +``` + +It's not hugely different from the default of 15 neighbors, but the clusters +are a bit more broken up. + +There should be a more pronounced difference going the other way and looking at +100 neighbors: + +```{r 100 neighbors} +set.seed(42) +iris_umap_nbrs100 <- umap(iris, n_neighbors = 100, min_dist = 0.3) +plot_umap(iris_umap_nbrs100) +``` + +Here there is a much more uniform appearance to the results. It's always worth +trying a few different values of `n_neighbors`, especially larger values, +although larger values of `n_neighbors` will lead to longer run times. Sometimes +small clusters that you think are meaningful may in fact be artifacts of setting +`n_neighbors` too small, so starting with a larger value and looking at the +effect of reducing `n_neighbors` can help you avoid over interpreting results. + +#### `init` + +The default initialization of UMAP is to use spectral initialization, which acts +upon the (symmetrized) k-nearest neighbor graph that is in determined by your +choice of `n_neighbors`. This is usually a good choice, but it involves a very +sparse matrix, which can sometimes be a bit *too* sparse, which leads to numerical +difficulties which manifest as slow run times or even hanging calculations. If +your dataset causes these issues, you can either try increasing `n_neighbors` +but I have seen cases where that would be inconvenient in terms of CPU and RAM +usage. An alternative is to use the first two principal components of the data, +which at least uses the data you provide to give a solid global picture of the +data that UMAP can refine. It's not appropriate for every dataset, but in most +cases, it's a perfectly good alternative. + +The only gotcha with it is that depending on the scaling of your data, the +initial coordinates can have large inter-point distances. UMAP will not optimize +that well, so such an output should be scaled to a small standard deviation. +If you set `init = "spca"`, it will do all that for you, although to be more +aligned with the UMAP coordinate initialization, I recommend you also set +`init_sdev = "range"` as well. `init_sdev` can also take a numerical value +for the standard deviation. Values from `1e-4` to `10` are reasonable, but I +recommend you stick to the default of `"range"`. + +```{r spca init} set.seed(42) -iris_umap_batch <- umap(iris, batch = TRUE, n_sgd_threads = 4) -# This will give the same results +iris_umap_spca <- + umap(iris, + init = "spca", + init_sdev = "range", + min_dist = 0.3) +plot_umap(iris_umap_spca) +``` + +This doesn't have a big effect on `iris`, but it's good to know about this as +an option: and it can also smooth out the effect of changing `n_neighbors` on +the initial coordinates with the standard spectral initialization, which can +make it easier to see the effect of changing `n_neighbors` on the final result. + +Some other `init` options to know about: + +* `"random"`: if the worst comes to the worst, you can always fall back to +randomly assigning the initial coordinates. You really want to avoid this if you +can though, because it will take longer to optimize the coordinates to the +same quality, so you will need to increase `n_epochs` to compensate. Even if you +do that, it's *much* more likely that you will end up in a minimum that is less +desirable than one based on a good initialization. This will make interpreting +the results harder, as you are more likely to end up with different clusters +beings split or mixed with each other. +* If you have some coordinates you like from another method, you can pass them +in as a matrix. But remember will probably want to scale them with `init_sdev` +though. + +#### `dens_scale` + +The `dens_scale` parameter varies from 0 to 1 and controls how much of the +relative densities of the input data is attempted to be preserved in the +output. + +```{r UMAP with density scaling} +set.seed(42) +iris_umapds <- umap(iris, min_dist = 0.3, dens_scale = 0.5) +plot_umap(iris_umapds) +``` + +This has shrunk the black cluster on the left of the plot (those are of species +`setosa`), which reflect that the density of the `setosa` points is less spread +out in the input data than the other two species. For more on `dens_scale` +please read its dedicated +[article](https://jlmelville.github.io/uwot/articles/leopold.html). + +## Embedding New Data + +Once you have an embedding, you can use it to embed new data, although you need +to remember to ask for a "model" to return. Instead of just the coordinates, you +will now get back a list which contains all the extra parameters you will need +for transforming new data. The coordinates are still available in the +`$embedding` component. + +Let's try building a UMAP with just the `setosa` and `versicolor` iris species: + +```{r create a UMAP model} set.seed(42) -iris_umap_batch2 <- umap(iris, batch = TRUE, n_sgd_threads = 2) -all(iris_umap_batch == iris_umap_batch2) -# TRUE -# Batch mode uses Adam optimizer by default. Control the parameters with the opt_args list -iris_umap_batch <- umap(iris, batch = TRUE, opt_args = list(beta1 = 0.9, beta2 = 0.999)) +iris_train <- iris[iris$Species %in% c("setosa", "versicolor"),] +iris_train_umap <- + umap(iris_train, min_dist = 0.3, ret_model = TRUE) +plot( + iris_train_umap$embedding, + col = iris_train$Species, + xlab = "", + ylab = "", + main = "UMAP setosa + versicolor" +) +``` + +Next, you can use `umap_transform` to embed the new points: + +```{r embed new coordinates} +iris_test <- iris[iris$Species == "virginica",] +set.seed(42) +iris_test_umap <- umap_transform(iris_test, iris_train_umap) +plot( + rbind(iris_train_umap$embedding, iris_test_umap), + col = iris$Species, + xlab = "", + ylab = "", + main = "UMAP transform virginica" +) ``` +The green points in the top-right show the embedded data. Note that the original +(black and red) clusters do not get optimized any further. While we haven't +perfectly reproduced the full UMAP, the `virginica` points are located in more +or less the right place, close to the `versicolor` items. Just like with any +machine learning method, you must be careful with how you choose your training +set. + ## Supported Distances For small (N < 4096) and Euclidean distance, exact nearest neighbors are found @@ -154,7 +321,13 @@ is the result of using the official Python UMAP implementation (via the [reticulate](https://cran.r-project.org/package=reticulate) package). Under that is the result of using `uwot`. -![mnist-py.png](articles/img/uwot/mnist-py.png) ![mnist-r.png](articles/img/uwot/mnist-r.png) +```{r, echo=FALSE, out.width="75%", fig.cap="MNIST UMAP (Python)"} +knitr::include_graphics("mnist-py.png") +``` + +```{r, echo=FALSE, out.width="75%", fig.cap="MNIST UMAP (R)"} +knitr::include_graphics("mnist-r.png") +``` The project documentation contains some more [examples](https://jlmelville.github.io/uwot/articles/umap-examples.html), and [comparison with Python](https://jlmelville.github.io/uwot/articles/pycompare.html).