From e4732ce77fab13631010c8c1828c230033d7a7df Mon Sep 17 00:00:00 2001 From: YPP Date: Tue, 19 Mar 2024 00:04:09 -0700 Subject: [PATCH] lect 18 --- lect18-scRNA-vae/Makefile | 12 + lect18-scRNA-vae/Setup.R | 231 +++ lect18-scRNA-vae/Util.R | 169 ++ lect18-scRNA-vae/lect18-scRNA-vae-beamer.pdf | Bin 0 -> 4145031 bytes lect18-scRNA-vae/lect18-scRNA-vae.Rmd | 1497 ++++++++++++++++++ lect18-scRNA-vae/torch_data.R | 55 + lect18-scRNA-vae/torch_etm_train.R | 121 ++ 7 files changed, 2085 insertions(+) create mode 100644 lect18-scRNA-vae/Makefile create mode 100644 lect18-scRNA-vae/Setup.R create mode 100644 lect18-scRNA-vae/Util.R create mode 100644 lect18-scRNA-vae/lect18-scRNA-vae-beamer.pdf create mode 100644 lect18-scRNA-vae/lect18-scRNA-vae.Rmd create mode 100644 lect18-scRNA-vae/torch_data.R create mode 100644 lect18-scRNA-vae/torch_etm_train.R diff --git a/lect18-scRNA-vae/Makefile b/lect18-scRNA-vae/Makefile new file mode 100644 index 0000000..7d7d5ed --- /dev/null +++ b/lect18-scRNA-vae/Makefile @@ -0,0 +1,12 @@ +RMD := $(wildcard *.rmd) $(wildcard *.Rmd) +PDF := $(RMD:.rmd=-slides.pdf) $(RMD:.Rmd=-beamer.pdf) +HTML := $(RMD:.rmd=.html) $(RMD:.Rmd=.html) +TEX := $(RMD:.rmd=.tex) $(RMD:.Rmd=.tex) + +all: $(PDF) + +clean: + rm -f $(PDF) + +%-beamer.pdf: %.Rmd + Rscript -e "rmarkdown::render('$<', output_format='beamer_presentation', output_file='$@');" diff --git a/lect18-scRNA-vae/Setup.R b/lect18-scRNA-vae/Setup.R new file mode 100644 index 0000000..47c6460 --- /dev/null +++ b/lect18-scRNA-vae/Setup.R @@ -0,0 +1,231 @@ +options(stringsAsFactors = FALSE) + +`%&%` <- function(a,b) paste0(a, b) +`%r%` <- function(mat,rr) mat[rr, , drop = FALSE] +`%c%` <- function(mat,cc) mat[, cc, drop = FALSE] + +library(tidyverse) +library(data.table) +library(ggrepel) + +num.int <- function(...) format(..., justify="none", big.mark=",", drop0trailing = TRUE) + +num.sci <- function(...) format(..., justify="none", digits=2, scientific = TRUE) + +row.order <- function(mat) { + require(cba) + require(proxy) + + if(nrow(mat) < 3) { + return(1:nrow(mat)) + } + + D = proxy::dist(mat, method <- function(a,b) 1 - cor(a,b, method = 'spearman')) + D[!is.finite(D)] = 0 + h.out = hclust(D) + o.out = cba::order.optimal(D, h.out$merge) + return(o.out$order) +} + +col.order <- function(pair.tab, .ro, ret.tab = FALSE) { + + M = pair.tab %>% + dplyr::select(row, col, weight) %>% + dplyr::mutate(row = factor(row, .ro)) %>% + tidyr::spread(key = col, value = weight, fill = 0) + + co = order(apply(M[, -1], 2, which.max), decreasing = TRUE) + .co = colnames(M)[-1][co] + if(ret.tab) { + ret = pair.tab %>% + dplyr::mutate(row = factor(row, .ro)) %>% + dplyr::mutate(col = factor(col, .co)) + } else { + ret = .co + } + return(ret) +} + +order.pair <- function(pair.tab, ret.tab=FALSE) { + + require(tidyr) + require(dplyr) + + .tab = pair.tab %>% dplyr::select(row, col, weight) + + M = .tab %>% tidyr::spread(key = col, value = weight, fill = 0) + rr = M[, 1] %>% unlist(use.names = FALSE) + cc = colnames(M)[-1] %>% unlist(use.names = FALSE) + + ## log.msg('Built the Mat: %d x %d', nrow(M), ncol(M)) + ro = row.order(M %>% dplyr::select(-row) %>% as.matrix()) + + ## log.msg('Sort the rows: %d', length(ro)) + co = order(apply(M[ro, -1], 2, which.max), decreasing = TRUE) + + ## co = row.order(t(M %>% dplyr::select(-row) %>% as.matrix())) + ## log.msg('Sort the columns: %d', length(co)) + + if(ret.tab){ + ret = pair.tab %>% + dplyr::mutate(row = factor(row, rr[ro])) %>% + dplyr::mutate(col = factor(col, cc[co])) + } else { + ret = list(rows = rr[ro], cols = cc[co], M = M) + } + + return(ret) +} + + +.sort.matrix <- function(.X) { + as.matrix(.X) %>% + reshape2::melt() %>% + rename(row = Var1, col = Var2, weight = value) %>% + order.pair(ret.tab=TRUE) %>% + as.data.table %>% + dcast(row ~ col, value.var = "weight") %>% + dplyr::select(-row) %>% + as.matrix +} + +.rnorm <- function(d1, d2) { + matrix(rnorm(d1 * d2), d1, d2) +} + +############################################################### +.matshow <- function(.mat, .lab=0, .axis.lab=0, .lw=0, .scale=TRUE) { + + library(ggrastr) + + .mat <- as.matrix(.mat) + .cols <- colnames(.mat) + .rows <- rownames(.mat) + if(length(.cols) < ncol(.mat)){ + colnames(.mat) <- str_c("c", 1:ncol(.mat)) + } + if(length(.rows) < nrow(.mat)){ + rownames(.mat) <- str_c("r", 1:nrow(.mat)) + } + .cols <- colnames(.mat) + .rows <- rownames(.mat) + + .dt <- + as.data.frame(.mat) %>% + rownames_to_column("row") %>% + as.data.table %>% + melt(id.vars = "row", variable.name = "col") %>% + dplyr::mutate(row = factor(as.character(row), rev(.rows))) %>% + dplyr::mutate(col = factor(as.character(col), .cols)) + + ret <- + ggplot(.dt, aes(y = row, x = col, fill = value)) + + theme(legend.position = "none") + if(.axis.lab > 0){ + ret <- + ret + + theme(axis.text.x = element_text(size=.axis.lab, angle=90, vjust=1, hjust=1)) + + theme(axis.text.y = element_text(size=.axis.lab)) + } else { + ret <- + ret + + theme(axis.text = element_blank()) + + theme(axis.ticks = element_blank()) + } + ret <- ret + + theme(axis.title = element_blank()) + + if(.lw > 0){ + ret <- ret + + ggrastr::rasterise(geom_tile(linewidth = .lw, colour = "black"), dpi=300) + } else { + ret <- ret + ggrastr::rasterise(geom_tile(), dpi=300) + } + + if(.scale){ + ret <- ret + + scale_fill_distiller(palette = "RdBu", direction = -1) + } else { + ret <- ret + + scale_fill_distiller(palette = "Greys", direction = 1) + } + + if(.lab > 0) { + ret <- ret + + geom_text(aes(label = round(value,1)), size = .lab) + } + + return(ret) +} + +################################################################ +.gg.plot <- function(...) { + ggplot(...) + + theme_classic() + + theme(axis.title = element_text(size=8)) + + theme(axis.text = element_text(size=6)) + + theme(legend.spacing = unit(.1, "lines"), + legend.key.size = unit(.5, "lines"), + legend.text = element_text(size=5), + legend.title = element_text(size=6), + panel.background = element_rect(fill='transparent'), + plot.background = element_rect(fill='transparent', color=NA), + legend.background = element_rect(fill='transparent', linewidth=0.05), + legend.box.background = element_rect(fill='transparent')) +} + +as.dt <- function(x, col.names=NULL) { + .mat <- as.matrix(x) + if(is.null(col.names)) col.names <- str_c("V",1:ncol(.mat)) + colnames(.mat) <- col.names + as.data.table(.mat) +} + +################################################################ +if.needed <- function(.file, .code) { + if(!all(file.exists(unlist(.file)))){ .code } + stopifnot(all(file.exists(unlist(.file)))) +} + +################################################################ +setup.env <- function(fig.dir) { + + ## save figures here + dir.create(fig.dir, showWarnings = FALSE, recursive = TRUE) + knitr::opts_chunk$set(warning = FALSE, message = FALSE, fig.path = fig.dir) + knitr::opts_chunk$set(echo=FALSE, fig.align="center") + + ## allow the code to chunk set size="tiny" ## + hook.chunk <- knitr::knit_hooks$get("chunk") + + ## Default: normalsize -> scriptsize + ## This will redefine normalsize + knitr::knit_hooks$set(chunk = function(x, options) { + x <- hook.chunk(x, options) + ifelse(options$size != "normalsize", + paste0("\n \\", options$size, "\n\n", x, "\n\n \\normalsize"), + paste0("\n \\", "scriptsize", "\n\n", x, "\n\n \\normalsize")) + }) + + ## show plot one by one in beamer ## + hook.plot <- knitr::knit_hooks$get("plot") + + knitr::knit_hooks$set(plot = function(x, options) { + if (!is.null(options$onslide.plot)) { + bf <- paste0("\\onslide<", options$onslide.plot, ">{") + ret <- paste(c(bf, knitr::hook_plot_tex(x, options), "}"), + collapse = "\n") + return(ret) + } else if (!is.null(options$only.plot)) { + bf <- paste0("\\only<", options$only.plot, ">{") + ret <- paste(c(bf, knitr::hook_plot_tex(x, options), "}"), + collapse = "\n") + return(ret) + } + return(hook.plot(x, options)) + }) + +## knitr::opts_chunk$set(tidy.opts = list(width.cutoff = 60), tidy = TRUE) + knitr::opts_chunk$set(dev = "cairo_pdf") +} + diff --git a/lect18-scRNA-vae/Util.R b/lect18-scRNA-vae/Util.R new file mode 100644 index 0000000..0726bb2 --- /dev/null +++ b/lect18-scRNA-vae/Util.R @@ -0,0 +1,169 @@ +sort.theta <- function(.dt){ + + .co <- order(apply(as.matrix(.dt[, -1]), 1, which.max)) + .to <- colnames(.dt)[-1] + + .melt <- .dt %>% + mutate(col = factor(1:n(), rev(.co))) %>% + melt(id.vars = c("col", "celltype")) %>% + mutate(variable = factor(variable, .to)) + + .cto <- + .melt[, + .(value = mean(value)), + by = .(celltype, variable)] %>% + dcast(celltype ~ variable, value.var = "value") %>% + (function(x){ + .mat <- as.matrix(x[,-1]) + .o <- order(apply(.mat, 1, which.max), decreasing = T) + rev(x$celltype[.o]) + }) + + .melt[, celltype := factor(celltype, .cto)] + return(.melt) +} + +read.theta <- function(.file){ + return(sort.theta(fread(.file))) +} + +plot.theta <- function(.file, with.ct = F){ + .dt <- read.theta(.file) + .thm <- + theme(axis.text.y = element_blank()) + + theme(axis.ticks.y = element_blank()) + + theme(legend.title = element_blank()) + + theme(legend.position = c(1,1)) + + theme(legend.justification = c(1,1)) + + ret <- + ggplot(.dt, aes(variable, col, fill = value)) + + ggrastr::rasterise(geom_tile(), dpi=300) + + scale_fill_distiller(direction=1) + + .thm + + if(with.ct){ + ret <- ret + + theme(strip.text.y = element_text(angle=0, hjust=0)) + + facet_grid(celltype ~ ., scales="free", space="free") + } + return(ret) +} + +################################################################ + +sort.beta <- function(.dt, genes.selected){ + + .genes <- rownames(.dt) + .mat <- as.matrix(.dt) + colnames(.mat) <- 1:ncol(.mat) + rownames(.mat) <- .genes + + .mat <- .mat[rownames(.mat) %in% genes.selected, , drop = F] + .melt <- reshape2::melt(.mat) %>% as.data.table() + .melt <- .melt[order(`value`, decreasing = T), head(.SD, 1), + by = .(Var1, Var2)] # remove redundancy + + .mat <- dcast(.melt, Var1 ~ Var2, value.var="value", fun.aggregate = max) + .rows <- unlist(.mat[,1], use.names = F) + .mat <- .mat[,-1] + + .ro <- apply(t(.mat), 2, which.max) %>% + order(decreasing = F) %>% + (function(x) .rows[x]) + .to <- colnames(.mat) + + .melt[, row := factor(`Var1`, .ro)] + .melt[, variable := factor(`Var2`, .to)] + + return(.melt) +} + +read.beta <- function(.file, ...){ + return(sort.beta(fread(.file), ...)) +} + +################################################################ +make.gs.lol <- function(.dt) { + .dt <- as.data.table(.dt) %>% unique() + .list <- + .dt[, .(gene = .(gene_symbol)), by = .(gs_name)] %>% + as.list() + .names <- .list$gs_name + .ret <- .list$gene + names(.ret) <- .names + return(.ret) +} + +run.beta.gsea <- function(.dt, .db){ + + vars <- unique(.dt$Var2) + ret <- data.table() + for(v in vars){ + + .scores <- as.vector(scale(.dt[Var2 == v]$value)) + names(.scores) <- .dt[Var2 == v]$Var1 + + .gsea <- fgsea::fgsea(pathways = .db, + stats = .scores, + scoreType = "pos") %>% + as.data.table() + .gsea[, Var2 := v] + ret <- rbind(ret, .gsea) + } + return(ret) +} + +read.panglao <- function(.organs = NULL, .file = "../data/PanglaoDB_markers_27_Mar_2020.tsv.gz"){ + + if(is.null(.organs)){ + .dt <- fread(.file) + } else { + .dt <- fread(.file)[organ %in% .organs] + } + + ret <- .dt %>% + rename(gene_symbol = `official gene symbol`, gs_name = `cell type`) %>% + mutate(gs_name = gsub(pattern="[ ]*cells", replacement = "", gs_name)) %>% + make.gs.lol() +} + +select.gsea.beta <- function(.gsea, .beta, ntop = 3, nmax = 30, sd.cutoff = 1e-4, padj.cutoff = .1){ + + ## cell types (pathways) to show + .cts <- unlist(.gsea[order(pval), + head(.SD, ntop), + by = .(Var2)][padj < padj.cutoff, + .(pathway)]) + + .gsea.show <- .gsea[pathway %in% .cts] + .gsea.show[, variable := factor(`Var2`, 0:100)] # 100 should be enough + + .temp <- dcast(.gsea.show, variable ~ pathway, + value.var = "pval", + fun.aggregate = min) + + .po <- order(apply(-log10(1e-20 + .temp[,-1]), 2, which.max)) + .po <- colnames(.temp)[-1][.po] + .gsea.show[, ct := factor(pathway, .po)] + + ## genes to show + .sd <- .beta[, .(s = sd(`value`)), by = .(Var1)] + + .genes.show <- intersect(unlist(.gsea.show$leadingEdge), + .sd[s > sd.cutoff]$Var1) + .beta.show <- .beta[Var1 %in% .genes.show] + + ## leading edges to show + .temp <- copy(.gsea.show) + .temp[, r := as.character(.I)] + .leading.edges <- .temp[, leadingEdge[[1]], by=r] %>% + left_join(.temp[, .(r, ct)]) %>% + mutate(row = factor(V1, sort(unique(.beta.show$row)))) + + .leading.genes <- .beta.show[row %in% .leading.edges$row][order(value, decreasing = T), head(.SD, 10), by = .(variable)][order(value, decreasing = T)]$row + .leading.genes <- head(unique(.leading.genes), nmax) + + list(gsea = .gsea.show, beta = .beta.show, leading.edges = .leading.edges, + leading.genes = .leading.genes) +} diff --git a/lect18-scRNA-vae/lect18-scRNA-vae-beamer.pdf 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zv&<+bB2YmHgk>nUE^$BCu{0WzGs|Lf?md(3Wm*?qxrA$1vuzS-Z!D*+D7@wrJlB$T zUN`=9h`%*9Qqi=v$Ma%8_R3w>3->NJ*X}6ff6NKPuH}{b^xx}|w9aEl>l6R6` z=!LM2V+(Ix{w=T?y3K1C80)(f5*cctwutYj|3N+=(6^F&iNfAwJAGbZW) literal 0 HcmV?d00001 diff --git a/lect18-scRNA-vae/lect18-scRNA-vae.Rmd b/lect18-scRNA-vae/lect18-scRNA-vae.Rmd new file mode 100644 index 0000000..1e04185 --- /dev/null +++ b/lect18-scRNA-vae/lect18-scRNA-vae.Rmd @@ -0,0 +1,1497 @@ +--- +title: "Statistical Methods for single cell data analysis 3" +author: "Yongjin Park, UBC Path + Stat, BC Cancer" +date: "`r format(Sys.time(), '%d %B %Y')`" +mainfont: Roboto +classoption: "aspectratio=169" +fontsize: 12pt +output: + powerpoint_presentation: + reference_doc: "_template.pptx" + html_document: + self_contained: true + beamer_presentation: + theme: Madrid + keep_md: true + keep_tex: true + latex_engine: xelatex + slide_level: 2 +header-includes: + - \usepackage{cancel} + - \definecolor{UBCblue}{rgb}{0.04706, 0.13725, 0.26667} + - \usecolortheme[named=UBCblue]{structure} + - \setbeamertemplate{frametitle}{\color{UBCblue}\bfseries\insertframetitle\par\vskip-6pt\hrulefill} + - \setbeamercolor{title in head/foot}{bg=white,fg=gray} + - \setbeamercolor{section in head/foot}{bg=white,fg=gray} + - \setbeamercolor{author in head/foot}{bg=white,fg=gray} + - \setbeamercolor{date in head/foot}{bg=white,fg=gray} + - \setbeamertemplate{page number in head/foot}{} + - \setbeamertemplate{frame numbering}[none] + - \setbeamercolor{alerted text}{bg=yellow} + - \AtBeginSection[]{\begin{frame}\frametitle{Today's lecture}{\Large\tableofcontents[currentsection]}\end{frame}} + - | + \makeatletter + \def\ps@titlepage{% + \setbeamertemplate{footline}{} + } + \addtobeamertemplate{title page}{\thispagestyle{titlepage}}{} + \makeatother + \include{toc} +--- + + +```{r setup, include=FALSE} +setwd("~/work/teaching/stat540_lectures/lect18-scRNA-vae/") +library(data.table) +library(tidyverse) +library(patchwork) +library(matrixTests) +library(mmutilR) +library(asapR) +source("Setup.R") +source("Util.R") +fig.dir <- "../Fig/scRNA_vae/" +dir.create(fig.dir, showWarnings=FALSE) +setup.env(fig.dir) +dir.create("../data", showWarnings=FALSE) +library(extrafont) +library(xkcd) +extrafont::font_import(pattern="[X/x]kcd", prompt=F) +extrafont::loadfonts() +theme_set(theme_xkcd() + + theme(title = element_text(size=10), + legend.background = element_blank()) + ) +``` + +## + +\large + +Source code available: + +`https://github.com/stat540-UBC/lectures` + +## Overview of single-cell data analysis + +\centerline{\includegraphics[width=\textwidth]{img/scRNA_overview/Karchenko_1.pdf}} + +\vfill + +\flushright +\tiny +Karchenko, \emph{Nature Methods} (2021) + +## Overview of single-cell data analysis cont'd + +\centerline{\includegraphics[width=\textwidth]{img/scRNA_overview/Karchenko_2.pdf}} + +\vfill + +\flushright +\tiny +Karchenko, \emph{Nature Methods} (2021) + +## The goal of modelling in high-dimensional space + +The goal is to model two types of relationships: + +\large + +1. Relationship between dimensions/features (genes) + +2. Relationship between samples/data points (cells) + +\normalsize + +\vfill + +* One of the most fundamental relationships is co-variation. + +## Covariance calculation in high-dimensional space + +Covariance between cell $i$ and $j$ across genes: + +$$\mathsf{cov}(X_{i}, X_{j}) = \mathbb{E}\!\left[X_{i} \, X_{j}\right] - \mathbb{E}\!\left[X_{i}\right] \mathbb{E}\!\left[X_{j}\right]$$ + +If $\mathbb{E}\!\left[X_{i}\right] = \mathbb{E}\!\left[X_{j}\right] = 0$ (e.g., standardization), +$$\mathsf{cov}(X_{i}, X_{j}) = \mathbb{E}\!\left[X_{i}\, X_{j}\right] \approx \frac{1}{p} \sum_{g=1}^{p} X_{gi} X_{gj}$$ + +## Covariance calculation in high-dimensional space - 2 + +Letting $\mathbf{x}_{i}^{\top} \equiv \left(X_{1i}, \ldots, X_{gi},\ldots, X_{pi} \right)$ and $\mathbb{E}\!\left[X_{gi}\right] \approx p^{-1} \sum_{g} X_{gi} = 0$ + +\onslide<2->{$$\mathsf{cov}(X_{i}, X_{j}) = \mathbb{E}\!\left[X_{i} X_{j}\right] +\approx \frac{1}{p} \sum_{g=1}^{p} X_{gi} X_{gj} = \frac{1}{p} \mathbf{x}_{i}^{\top} \mathbf{x}_{j}$$} +\onslide<3>{ +Full sample covariance: +$$\frac{1}{p} X^{\top} X = +\frac{1}{p} +\left( +\begin{array}{l l l} +\mathbf{x}_{1}^{\top} \mathbf{x}_{1} & \cdots & \mathbf{x}_{1}^{\top}\mathbf{x}_{n} \\ +\mathbf{x}_{2}^{\top}\mathbf{x}_{1} & \cdots & \mathbf{x}_{2}^{\top}\mathbf{x}_{n} \\ +& \vdots & \\ +\mathbf{x}_{n}^{\top}\mathbf{x}_{1} & \cdots & \mathbf{x}_{n}^{\top}\mathbf{x}_{n} +\end{array} +\right)$$} + +## Singular value decomposition simplifies covariance + +$$X = U D V^{\top}$$ + +\only<1>{ +Gene covarince +\begin{eqnarray*} +XX^{\top} +&=& U D V^{\top} (U D V^{\top})^{\top} \\ +&=& U D V^{\top} ((V^{\top})^{\top} D^{\top} U^{\top}) \\ +&=& U D \cancelto{I}{V^{\top} V} D U^{\top} \\ +&=& U D^{2} U^{\top} +\end{eqnarray*}} + +\only<2>{ +Sample covarince +\begin{eqnarray*} +X^{\top} X +&=& (U D V^{\top})^{\top} U D V^{\top} \\ +&=& ((V^{\top})^{\top} D^{\top} U^{\top}) U D V^{\top} \\ +&=& V D \cancelto{I}{U^{\top} U} D V^{\top} \\ +&=& V D^{2} V^{\top} +\end{eqnarray*}} + +* The Matrix Cookbook: [https://www2.imm.dtu.dk/pubdb/pubs/3274-full.html](https://www2.imm.dtu.dk/pubdb/pubs/3274-full.html) + + +## Example: scRNA-seq data of human pancreatic cells + +```{r} +raw.data <- fileset.list("../data/GSE85241/GSE85241") +.info <- rcpp_mmutil_info(raw.data$mtx) +``` + +:::::: {.columns} +::: {.column width=.45} + +\centerline{\includegraphics[height=.7\textheight]{img/Muraro_GSE85241.png}} + +::: +::: {.column width=.45} + +\large + +* `r num.int(.info$max.row)` genes/features/rows + +* `r num.int(.info$max.col)` cells/columns + +* `r num.int(.info$max.elem)` non-zero elements + +* $\approx$ `r round(100 * .info$max.elem / .info$max.col / .info$max.row)` % non-zero elements + +::: +:::::: + +\tiny +Muraro \textit{et al.} \emph{Cell Systems} (2016) + + +# Recent developments in latent factor modelling + +## Recap: How can we learn patterns in unsupervised learning from single cell count data? + +\centerline{\includegraphics[width=.75\linewidth]{img/scRNA_ETM/Matrix_Factorization.pdf}} + +* Goal: Find the factor-specific gene dictionary $\beta$ and hidden factor loading $H$. + +## Can we "encode" high-dim data to low-dim hidden variables? + +:::::: {.columns} +::: {.column width=.5} + +\only<1>{\centerline{\includegraphics[width=.85\linewidth]{img/scRNA_ETM/encoder_1.pdf}}} + +\only<2>{\centerline{\includegraphics[width=\linewidth]{img/scRNA_ETM/encoder_deep1.pdf}}} + +::: +::: {.column width=.5} + +* Take one cell as a long vector $\mathbf{x}_{i}$ + +* Apply an encoding function $f(\mathbf{x}_{i})$ + +* Neural network architectures capture relationships between variables + +\onslide<2>{ +$$h_{i}^{(l)} \gets f(\sum_{j} \underset{\textsf{\color{magenta} connections}}{W_{ij}} h_{j}^{(l-1)} + \underset{\textsf{\color{teal} bias}}{b_{i}})$$ +} + +* no connection within each layer + +::: +:::::: + +## + +\Large + +Unsupervised learning of a good encoder model is fundamentally challenging because ... + +\vfill + +\onslide<2>{we don't have good encoding ``examples'' beforehand.} + +## How can we "supervise" how well we encoded? + +\centerline{\includegraphics[width=.6\linewidth]{img/scRNA_ETM/autoencoder_with_nodes.pdf}} + +## "Supervise" unsupervised learning by self reconstruction + +\centerline{\includegraphics[width=.5\linewidth]{img/scRNA_ETM/autoencoder.pdf}} + + +## Deep autoencoder first proposed in computer vision + +:::::: {.columns} +::: {.column width=.6} + +\centerline{\includegraphics[width=.9\linewidth]{img/scRNA_ETM/Autoencoder_DL.pdf}} + +::: +::: {.column width=.4} + +* [Hinton & Salakhutdinov, Science (2006)](https://www.science.org/doi/10.1126/science.1127647) + +* One of the first "deep learning" idea + +* Learning the latent representation of an image helps subsequent classification tasks. + +* To train a deep autoencoder architecture, they pretrained the model layer by layer. + +::: +:::::: + +## Let's build a simple autoencoder model using `torch` + +```{r eval=F, echo = T} +############################################################### +## R wrapper library for `libtorch` (C++ engine for PyTorch) ## +############################################################### + +library(torch) + +############################################ +## We can use GPU (NVIDIA Cuda), optional ## +############################################ + +GPU <- torch_device("cuda") +``` + +* See this online book if you are interested in building your own deep learning model: **Deep Learning and Scientific Computing with R torch** + +\small + +* [https://skeydan.github.io/Deep-Learning-and-Scientific-Computing-with-R-torch/](https://skeydan.github.io/Deep-Learning-and-Scientific-Computing-with-R-torch/) + +```{r read_full_data} +library(torch) +GPU <- torch_device("cuda") + +read.full.data <- function(.data, .dev = GPU){ + + .info <- rcpp_mmutil_info(.data$mtx) + .idx <- rcpp_mmutil_read_index(.data$idx) + .mtx <- .data$mtx + .out <- mmutilR::rcpp_mmutil_read_columns_sparse(.mtx, .idx, 1:.info$max.col) + + .inds <- rbind(.out$col, .out$row) # column is row & vice versa + .dims <- c(.out$max.col, .out$max.row) # dimensionality + + xx <- torch_sparse_coo_tensor(indices = .inds, + values = .out$val, + size = .dims, + dtype = torch_float(), + device = .dev) + + return(xx$to_dense()$to(device=.dev)) +} + +raw.data <- fileset.list("../data/GSE85241/GSE85241") +x.torch <- read.full.data(raw.data) +``` + +## Why do we use some Deep Learning library? + +\large + +* Built-in functions for scientific computing + + * `log`, `log1p`, `exp`, `softmax`, `sigmoid`, `softplus` + +* Faster computation both in CPU and GPU (in general) + +* **No need to work on differentiation** by hands + +* Trouble shooting by ML community + +## Encoder: high-dim data $\to$ low-dim latent space + +```{r echo = T} +build.linear.encoder <- + nn_module( + ## How do we want to build this module ## + classname = "linear encoder", + initialize = function(d.in, K){ + self$K <- K # number of hidden factors + self$func.z <- nn_linear(d.in, K) # a linear neural net layer + self$bn <- nn_batch_norm1d(d.in) # batch norm + }, + ## Define how do we propate infomration ## + forward = function(x.b){ + self$get.latent(x.b) + }, + get.latent = function(x.b){ + x.b <- torch_log1p(x.b) # log1p transformation + x.b <- self$bn(x.b) # to expedite training + self$func.z(x.b) # take linear combinations + }) +``` + +## Check if this encoder module works okay + +```{r echo = T} +lin.encoder <- build.linear.encoder(ncol(x.torch), 7)$to(dev = GPU) +lin.encoder + +########################################## +## Test using the first five cells/data ## +########################################## + +lin.encoder(x.torch[1:5, ]) +``` + +## Decoder: low-dim latent space $\to$ high-dim data + +```{r echo=T} +build.linear.decoder <- + nn_module( + classname = "linear decoder", + ## Define how we want to build this module + initialize = function(n.out, K) { + self$K <- K + self$func.logX <- nn_linear(K, n.out) # latent dim -> data dim + }, + ## Define how do get back high-dim data + forward = function(z.b){ + .num <- self$func.logX(z.b) + .denom <- torch_logsumexp(.num, dim = -1, keepdim = T) + return(.num - .denom) + }, + ## Helper function + get.weight = function(){ + self$func.logX$weight + }) +``` + +## Check flow from the encoder to decoder + +```{r echo = T} +lin.decoder <- build.linear.decoder(ncol(x.torch), K=7)$to(device=GPU) + +x.input <- x.torch[1:5, ] +z.b <- lin.encoder(x.input) + +############################################# +## reconstruction of x based on the latent ## +############################################# +logx.recon <- lin.decoder(z.b) +logx.recon[, 1:7] +``` + +* Note: the reconstructed data matrix is in logarithm scale + +## What's really going on in terms of functions? + +For each sample/cell $i$, + +* Encoder: + +$$\mathbf{z}_{i} \gets \log(\mathsf{normalize}(\mathbf{x}_{i}) + 1) \cdot \mathbf{W}^{(z)} + \mathbf{b}^{(z)}$$ + +* Decoder: + +$$\hat{\mathbf{x}}_{i} \gets \mathbf{z}_{i} \cdot \mathbf{W}^{(x)} + \mathbf{b}^{(x)}$$ + + +## Goal: to make the reconstructed data $\approx$ the input + +* Gene expression frequency: +$$\rho_{ig} = \frac{\exp( \widehat{{\log}X}_{ig} )}{\sum_{g'} \exp( \widehat{{\log}X}_{ig'} )}$$ + +* Multinomial log-likelihood: +$$L_{i} \overset{\mathsf{def}}{=} \sum_{g=1}^{p} X_{ig} \log \rho_{ig}$$ + +\only<1>{$$\log \rho_{ig} = \widehat{{\log}X}_{ig} - \log \sum_{g'} \exp(\widehat{{\log}X}_{ig'})$$} + +\only<2>{$$L_{i} = \sum_{g=1}^{p} X_{ig} \left[ \widehat{\log X}_{ig} - \mathsf{logSumExp}(\widehat{\log \mathbf{x}}_{i}) \right]$$} + +## Goal: to make the reconstructed data $\approx$ the input + +:::::: {.columns} +::: {.column width=.6} + +```{r echo = T} +multinom.llik <- function(x.input, logx.recon){ + torch_sum(x.input * logx.recon, dim = -1) +} + +multinom.llik(x.input, logx.recon) +``` + +::: +::: {.column width=.4} + +* We need to maximize this with respect to all the parameters in the encoder and decoder layers. + +* Maximizing log-likelihood $\iff$ minimizing *negative* log-likelihood + +::: +:::::: + +## `Torch` provides a convenient way to "differentiate" + +```{r echo = T, size="large"} +loss <- -torch_mean(multinom.llik(x.input, logx.recon)) +loss +loss$backward() +``` + +## Put the encoder and decoder together (so that gradients flow) + +```{r echo = T} +build.linear.autoenc <- + nn_module( + classname = "linear autoencoder", + initialize = function(d.data, K){ + self$enc <- build.linear.encoder(d.data, K) + self$dec <- build.linear.decoder(d.data, K) + }, + forward = function(x){ + z <- self$enc(x) + x.hat <- self$dec(z) + .loss <- - multinom.llik(x, x.hat) + list(loss = .loss) + }) +``` + +```{r} +linear.autoenc <- build.linear.autoenc(ncol(x.torch), K=12) +linear.autoenc$to(device=GPU) +``` + +## A bit more formal definition + +Minimize the total loss + +$$L \overset{\textsf{def}}{=} \sum_{i=1}^{n}\mathsf{Loss}(\mathbf{x}_{i}, \widehat{\mathbf{x}}_{i})$$ + +where + +$$\hat{\mathbf{x}}_{i} = \mathsf{Decoder}(\mathsf{Encoder}(\mathbf{x}_{i}; \theta^{(\textsf{enc})}); \theta^{(\textsf{dec})})$$ + +with respect to the parameters $\theta$. + +## Update by stochastic gradient steps + +Take gradient (direction to minimize the loss) for each parameter: + +$$\nabla L \equiv \left( \frac{\partial L}{\partial \theta_{1}}, \frac{\partial L}{\partial \theta_{2}}, \ldots \right)$$ + +Update parameters: + +$$\theta^{(t)} \gets \underset{\textsf{\color{blue} the previous}}{\theta^{(t-1)}} - \underset{\textsf{\color{teal} learning rate}}{\rho_{t}} \underbrace{\nabla L}_{\textsf{\color{magenta} gradient at $t-1$}}$$ + +## Update by stochastic gradient steps + +```{r echo = T} +## register parameters to ADAM optimizer +.params <- linear.autoenc$parameters +adam <- optim_adam(.params, lr = 1e-2) +adam$zero_grad() +``` + +:::::: {.columns} +::: {.column width=.5} + +```{r echo = T} +x.b <- x.torch[1:3, ] # Select minibatch data +out <- linear.autoenc(x.b) # data -> latent -> recon. +out$loss # loss evalulated on X.b +``` + +::: +::: {.column width=.5} + +1. Take minibatch $X^{(b)}$ + +2. Recon. $\hat{X}^{(b)} = \mathsf{Dec}(\mathsf{Enc}(X^{(b)}))$ + +3. Evaluate $\mathsf{Loss}(X^{(b)}, \hat{X}^{(b)})$ + +::: +:::::: + +## Update by stochastic gradient steps + +:::::: {.columns} +::: {.column width=.6} + +```{r echo = T} +loss.b <- torch_sum(out$loss) # +loss.b$backward() # numerically + # differentiate + +## Before we take one SGD step +head(adam$param_groups[[1]]$params$enc.func.z.bias, 2) +adam$step() -> . +## After we take one SGD step +head(adam$param_groups[[1]]$params$enc.func.z.bias, 2) +``` + +::: +::: {.column width=.4} + +\onslide<2->{$\star$ Aggregate training loss across samples in this minibatch: + +$\sum_{i \in \mathsf{minibatch}(b)} \mathsf{loss}(\mathbf{x}_{i}, \hat{\mathbf{x}_{i}})$} + +${}$ + +\onslide<3->{$\star$ Take gradient with respect to encoder and decoder parameters} + +${}$ + +\onslide<4>{$\star$ Update the parameters by taking one (stochastic) gradient descent step + +$\theta^{(t)} \gets \theta^{(t-1)} + \rho \nabla L$} + +::: +:::::: + + + +## Training algorithm: Repeat SGD steps many times + +\large + +* For many epochs + + 1. Sample mini batch data + + 2. Evaluate loss function $L(\mathbf{x}_{i}, \hat{\mathbf{x}}_{i})$ + + 3. Compute gradient $\nabla_{\theta} L$ + + 4. Update parameters by SGD + +* Report encoding results + +```{r include = F} +train.model <- function(xx, model, rds.file, + max.epoch = 200, + batch.size = 128, + learning.rate = 1e-3, + record = 10){ + + GPU <- torch_device("cuda") + CPU <- torch_device("cpu") + + adam <- optim_adam(model$parameters, lr = learning.rate) + + loss.vec <- c() + + ntot <- nrow(xx) + nbatch <- ceiling(ntot /batch.size) + nn <- batch.size * nbatch + + hh.list <- list() + hh <- matrix(NA, nrow = ntot, ncol = model$enc$K) + + ww.list <- list() + ww <- matrix(NA, nrow = ncol(xx), ncol = model$enc$K) + + for(tt in seq(0, max.epoch)){ + .shuffle <- sample(ntot, nn) + .loss.tot <- 0 + + for(b in seq(0, nbatch-1)){ + .idx <- seq(1 + b * batch.size, (b+1) * batch.size) + .idx <- .shuffle[.idx] + xx.b <- xx[.idx, ] + + model$train() + adam$zero_grad() + out <- model(xx.b) + loss.b <- torch_sum(out$loss)$to(device=CPU)$item() + torch_mean(out$loss)$backward() + adam$step() -> . + .loss.tot <- .loss.tot + loss.b + + cat(tt, " ", loss.b, "\r", file = stderr()) + flush(stderr()) + } + + loss.vec <- c(loss.vec, .loss.tot/nn) + + if(tt %% record == 0){ + model$eval() + + for(b in seq(0, nbatch-1)){ + .idx <- seq(1 + b * batch.size, min(ntot, (b+1) * batch.size)) + xx.b <- xx[.idx, ] + hh.b <- model$enc$get.latent(xx.b)$to(device=CPU) + hh[.idx, ] <- as.matrix(hh.b) + } + + ww <- as.matrix(model$dec$get.weight()$to(device=CPU)) + hh.list <- c(hh.list, list(hh)) + ww.list <- c(ww.list, list(ww)) + } + } + saveRDS(list(loss = loss.vec, latent = hh.list, weights = ww.list), file = rds.file) +} +``` + +```{r run_linear_autoencoder} +.file <- fig.dir %&% "/linear_autoencodeer.rds" +if.needed(.file, { + train.model(x.torch, linear.autoenc, .file, max.epoch = 1000, record=25) +}) +``` + +## Results: SGD minimized the loss function + +```{r} +.result <- readRDS(.file) +``` + +```{r fig.width=5, fig.height = 3} +nepoch <- length(.result$loss) +.df <- data.frame(epoch = seq(0, nepoch - 1), loss = .result$loss) +ggplot(.df, aes(epoch, loss)) + + geom_point(stroke = 0) +``` + +## Hidden representations (factor $\times$ cell) + +```{r fig.width=5, fig.height=2.5, only.plot="1"} +tt <- 1 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="2"} +tt <- 2 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="3"} +tt <- 10 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="4"} +tt <- 20 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="5"} +tt <- 30 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="6"} +tt <- 40 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +## Latent dimensions estimated by the encoder model + +```{r} +tt <- 41 +.df <- as.data.frame(.result$latent[[tt]]) +``` + +```{r fig.width=6, fig.height=2.2, only.plot="1"} +p1 <- ggplot(.df, aes(V1, V2)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p2 <- ggplot(.df, aes(V3, V4)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p3 <- ggplot(.df, aes(V5, V6)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p1 | p2 | p3 +``` + +```{r fig.width=6, fig.height=2.2, only.plot="2"} +p1 <- ggplot(.df, aes(V7, V8)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p2 <- ggplot(.df, aes(V9, V10)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p3 <- ggplot(.df, aes(V11, V12)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p1 | p2 | p3 +``` + +## Variational autoencoder (VAE) + +:::::: {.columns} +::: {.column width=.5} + +\centerline{A classical autoencoder:} + +${}$ + +\centerline{\includegraphics[width=\linewidth]{img/scRNA_ETM/autoencoder.pdf}} + +::: +::: {.column width=.5} + +\centerline{Variational autoencoder:} + +${}$ + +\centerline{\includegraphics[width=\linewidth]{img/scRNA_ETM/VAE.pdf}} + +::: +:::::: + +\vfill + +* Define relationships between variables (auto generative process) + +* Usually, the decoder side captures our scientific hypothesis + + + +## VAE approximates Bayesian inference + +:::::: {.columns} +::: {.column width=.5} + +\centerline{\includegraphics[width=\linewidth]{img/scRNA_ETM/VAE_paper.pdf}} + +::: +::: {.column width=.5} + +\large + +Prior: + +$$Z \sim p(Z| \psi)$$ + +Data likelihood: + +$$X \sim p(X|Z, \theta)$$ + +Variational Encoder: + +$$q(Z|X) = \mathsf{NN}(X)$$ + +::: +:::::: + +## SCVI: deep generative model for scRNA-seq + +\vfill + +\centerline{\includegraphics[height=.65\textheight]{img/scRNA_ETM/scVI.pdf}} + +\vfill + +Generative model: zero-inflated negative binomial distribution + +\vfill +\tiny + +Lopez, .., Jordan, Yosef, \emph{Nature Methods} (2018) + +## A new encoder as a posterior inference machine + +\centerline{\includegraphics[width=.45\linewidth]{img/scRNA_ETM/VAE.pdf}} + +We need to define functions (neural networks) that maps from the data vector $\mathbf{x}$ to + +* the mean of latent embedding: $\mu$ + +* the log variance of latent embedding: $\sigma$ + +* $z_{ig} \gets \mu_{ig} + \sigma_{ig} \epsilon,\quad \epsilon \sim \mathcal{N}\!\left(0, 1\right)$ + +## A new encoder as a posterior inference machine + +```{r echo = T} +build.vae.encoder <- nn_module( + classname = "vae encoder", + initialize = function(d.in, K){ + self$K <- K # number of hidden vars. + self$z.mean <- nn_linear(d.in, K) # mean function + self$z.logvar <- nn_linear(d.in, K) # log variance function + self$bn <- nn_batch_norm1d(d.in) # batch norm + }, + forward = function(x.b){ + x.b <- self$normalize(x.b) + mm <- self$z.mean(x.b) # mean evaluated + lv <- torch_clamp(self$z.logvar(x.b), -4.0, 4.0) # log-var evaluated + z <- mm + torch_randn_like(lv) * torch_exp(lv/ 2.) # stochastic z + list(z = z, z.mean = mm, z.logvar = lv) + }, + normalize = function(x.b){ # normalization + x.b <- torch_log1p(x.b) # log1p transformation + self$bn(x.b) # to expedite training + }, ## helper function + get.latent = function(x.b){ self$z.mean(self$normalize(x.b)) }) +``` + +## We will have two types of loss functions + +:::::: {.columns} +::: {.column width=.5} + +Data log likelihood (a generative model) + +$$\log \prod_{i=1}^{n} \underset{\textsf{\color{teal} multinomial}}{p(\mathbf{x}_{i}|\mathbf{z}_{i})}$$ + +```{r echo = T} +multinom.llik <- function(x.input, logx.recon){ + torch_sum(x.input * logx.recon, dim = -1) +} +``` + +::: +::: {.column width=.5} + +Divergence between prior and posterior + +$$D_{\mathsf{KL}}\left(\underset{\textsf{\color{teal} posterior}}{q(\mathbf{z})} \middle\| \underset{\textsf{\color{magenta} prior}}{p(\mathbf{z})}\right)$$ + +```{r echo = T} +kl.loss <- function(.mean, .lnvar) { + -0.5 * torch_sum(1. + .lnvar + - torch_pow(.mean, 2.) + - torch_exp(.lnvar), + dim = -1) +} +``` + +::: +:::::: + +:::::: {.columns} +::: {.column width=.5} + +* The log-likelihood is the same as before + +* KL loss will work like regularization + +::: +::: {.column width=.5} + +* We assume both $q$ and $p$ follows Gaussian distribution + +::: +:::::: + +## A complete definition of our VAE model + +```{r echo = T} +build.vae <- + nn_module( + classname = "variational autoencoder", + initialize = function(d.data, K){ + self$enc <- build.vae.encoder(d.data, K) # encoder model + self$dec <- build.linear.decoder(d.data, K) # decoder model + }, + forward = function(x){ + .enc <- self$enc(x) # + x.hat <- self$dec(.enc$z) # reconstruction + .llik <- multinom.llik(x, x.hat) # data likelihood + .kl <- kl.loss(.enc$z.mean, .enc$z.logvar) # KL divergence + .loss <- .kl - .llik # combined loss + list(loss = .loss, kl=.kl) + } + ) +``` + +* What do you want to change? + +## VAE: Check flow from the encoder to decoder + +:::::: {.columns} +::: {.column width=.5} + +```{r echo = T} +vae <- build.vae(ncol(x.torch), K=12) +vae$to(device = GPU) + +x.input <- x.torch[1:5, ] +out <- vae(x.input) +out$loss +``` + +::: +::: {.column width=.5} + +```{r echo = T} +############################################# +## reconstruction of x based on the latent ## +############################################# +z.b <- vae$enc(x.input) +logx.recon <- vae$dec(z.b$z) +logx.recon[, 1:5] +``` + +::: +:::::: + +* Note: the reconstructed data matrix is in logarithm scale + +```{r} +.file <- fig.dir %&% "/vae.rds" +if.needed(.file, { + train.model(x.torch, vae, .file, max.epoch = 1000, record=21) +}) +``` + +## Results: SGD minimized the VAE loss function + +```{r} +.result <- readRDS(.file) +``` + +```{r fig.width=5, fig.height = 3} +nepoch <- length(.result$loss) +.df <- data.frame(epoch = seq(0, nepoch - 1), loss = .result$loss) +ggplot(.df, aes(epoch, loss)) + + geom_point(stroke = 0) +``` + +## Latent dimensions in the VAE model + +```{r} +tt <- 41 +.df <- as.data.frame(.result$latent[[tt]]) +``` + +```{r fig.width=6, fig.height=2.2, only.plot="1"} +p1 <- ggplot(.df, aes(V1, V2)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p2 <- ggplot(.df, aes(V3, V4)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p3 <- ggplot(.df, aes(V5, V6)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p1 | p2 | p3 +``` + +```{r fig.width=6, fig.height=2.2, only.plot="2"} +p1 <- ggplot(.df, aes(V7, V8)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p2 <- ggplot(.df, aes(V9, V10)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p3 <- ggplot(.df, aes(V11, V12)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p1 | p2 | p3 +``` + +## Hidden representations (factor $\times$ cell) + +```{r fig.width=5, fig.height=2.5, only.plot="1"} +tt <- 1 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="2"} +tt <- 2 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="3"} +tt <- 10 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="4"} +tt <- 20 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="5"} +tt <- 30 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="6"} +tt <- 40 +.temp <- apply(.result$latent[[tt]], 2, scale) +.matshow(.sort.matrix(t(.temp))) + ggtitle("epoch = " %&% (1 + (tt - 1) * 25)) +``` + +## Annotate factors to cell types by enrichment (`fgsea`) + +```{r run_gsea_beta} +.beta <- apply(.result$weights[[tt]], 2, scale) +.rows <- fread(raw.data$row, header=F) +.rows[, c("gene","chr") := tstrsplit(`V1`, split="[_]+", keep=1:2)] +rownames(.beta) <- .rows$gene + +.db <- read.panglao("Pancreas") +.genes <- unique(unlist(.db)) +.dt <- sort.beta(.beta, genes.selected = .genes) +.gsea <- run.beta.gsea(.dt, .db) +.show <- select.gsea.beta(.gsea, .dt, sd.cutoff = 1e-4) +``` + +```{r} +p1 <- + .gg.plot(.show$beta, aes(row, variable, fill=`value`)) + + theme(axis.text.x = element_blank()) + + theme(axis.ticks.x = element_blank()) + + theme(legend.position = "none") + + ggrastr::rasterise(geom_tile(), dpi=300) + + ylab("VAE factors") + xlab("genes") + + scale_fill_distiller(direction=-1, palette="RdBu") + +p2 <- + .gg.plot(.show$leading.edges, aes(row, ct)) + + theme(axis.text.x = element_blank()) + + theme(axis.ticks.x = element_blank()) + + theme(panel.grid = element_blank()) + + geom_tile(linewidth=0) + + ylab("cell types") + +p3 <- + .gg.plot(.show$gsea, aes(variable, ct, fill=-log10(pmax(padj, 1e-4)))) + + xlab("NMF factors") + ylab("") + + scale_y_discrete(position = "right") + + geom_tile(linewidth=.1, color="black") + + theme(legend.key.width=unit(.2, "lines")) + + theme(legend.text = element_text(size=4)) + + scale_fill_distiller("Padj", palette = "YlGnBu", direction=1, labels=function(x) num.sci(10^(-x))) +``` + +```{r fig.width=6, fig.height=3, only.plot="1"} +print(p1/p2) +``` + +```{r fig.width=6, fig.height=3, only.plot="2"} +C <- length(unique(.show$gsea$ct)) +K <- length(unique(.show$beta$variable)) +p0 <- ggplot() + theme_void() +wrap_plots(p1, p0, p2, p3, nrow = 2, heights=c(K,C), widths=c(4,1)) +``` + +## + +\large + +Can we improve model interpretation? + +\begin{enumerate} +\item<2-> The decoder part is open to modelling in many different ways. +\item<3> Can we define latent factors by gene expression frequencies? +\end{enumerate} + +## Single-cell Embedded Topic Model + +\centerline{\includegraphics[width=\linewidth]{img/scRNA_ETM/scETM.pdf}} + +\vfill + +Zhao .. Li, Nature Comm. (2021) + +## Document topic modelling + +\centerline{\includegraphics[width=.8\linewidth]{img/scRNA_ETM/BleiTopicLDA_0.pdf}} + +\vfill +\tiny +\flushright +Slide credit: David Blei + +## Word frequencies define topics in documents + +\centerline{\includegraphics[width=.8\linewidth]{img/scRNA_ETM/BleiTopicLDA_1.pdf}} + +\vfill +\tiny +\flushright +Slide credit: David Blei + +## Multinomial topic model for scRNA-seq data + +Can we simply model scRNA-seq counts by multinomial distribution? + +\only<1>{ +\centerline{\includegraphics[height=.4\textheight]{img/scRNA_ETM/ETM.pdf}} +} + +\vfill + +\only<2>{ +Likelihood model: + +$$\mathcal{L} = \prod_{i=1}^{n}\prod_{g=1}^{\textsf{genes}} \left(\sum_{k} H_{ik} \beta_{kg}\right)^{X_{ij}}$$ +} + +\only<3>{ +Likelihood model: + +$$\mathcal{L} = \prod_{i=1}^{n}\prod_{g=1}^{\textsf{genes}} \left(\underset{\textsf{\color{magenta} a gene $g$'s probability in a cell $i$} \equiv \rho_{ig}}{\sum_{k} H_{ik} \beta_{kg}}\right)^{X_{ij}}$$} + +\normalsize + +* $X_{ig}$: gene expression of a gene $g$ in a single cell $i$ + +* $H_{ik}$: latent topic proportion of a cell $i$ to a topic $k$ + +* $\beta_{kg}$: topic $k$-specific gene probability + + +## Topic modelling for single-cell data + +\vfill + +\centerline{\includegraphics[height=.4\textheight]{img/scRNA_ETM/ETM.pdf}} + +\vfill + +:::::: {.columns} +::: {.column width=.5} + +Probability of gene $g$ in a cell $i$: + +$$\rho_{ig} = \sum_{k \in \textsf{topics}} H_{ik} \beta_{kg}$$ + +::: +::: {.column width=.45} + +By **not** normalizing the probability of each cell, we do not worry about modelling sequencing depths. + +::: +:::::: + + +## Document topic modelling vs. single-cell ETM + +```{r echo = FALSE, results="asis"} +.dt <- data.table( + variables = c("$D$", + "$d$", + "$N_{d}$", + "$j$", + "$K$", + "$k$", + "$V$", + "$v$", + "$W_{dj}^{v}$", + "$X_{dv}$"), + `in document topic model` = c("Total number of documents (corpus)", + "Document index", + "Number of words in a document $d$", + "Word index, $j \\in [N_{d}]$", + "Total number of topics", + "Topic index, $k \\in [K]$", + "Size of vocabulary", + "Vocabulary index $v \\in [V]$", + "Indicator for a word to vocabulary $\\in \\{0, 1\\}$", + "Vocabulary $v$ occurrence in a document $d$"), + `in single cell ETM` = c("Total number of cells", + "Cell index", + "Number of read counts in a cell $d$", + "Read index", + "Total number of cell type topics", + "Cell topic index", + "Total number of genes", + "Gene index", + "Indicator for a read to a gene $\\in \\{0, 1\\}$", + "Gene expression of a gene $v$ in a cell $d$ $\\in [0, N_{d}]$")) + +knitr::kable(.dt) +``` + + +## + +```{r echo = FALSE, results = "asis"} +.dt.param <- data.table( + variables = c("$Z_{dj}^{k}$", + "$H_{dk}$", + "$\\beta_{kv}$"), + `in document topic model` = c("Indicator for assigning a word to a topic $k$", + "Hidden state $k$ of a document $d$", + "topic $k$-specific vocabulary $v$ frequency"), + `in single cell ETM` = c("Indicator for assigning a read to a topic $k$", + "Hidden state $k$ of a cell $d$", + "topic $k$-specific, a gene $v$'s exression")) + +knitr::kable(.dt.param) +``` + +* In Latent Dirichlet Allocation: $\sum_{k=1}^{K} H_{dk} = 1$ and $H_{dk} > 0$, and $\mathbf{h}_{d} \sim \mathsf{Dirichlet}(\alpha/K, \ldots, \alpha/K)$ _a priori_. Approximately, we have $\hat{H}_{dk} = \sum_{j}^{N_{d}} Z_{dj}^{k} / N_{d}$. + +* In Embedded Topic model, $H_{dk}$ with the simplex constraints; $H_{dk} = \exp(\delta_{dk}) / \sum_{k'} \exp(\delta_{dk'})$ where $\delta_{dk} \sim \mathcal{N}\!\left(0,1\right)$ _a priori_. + +* Additional constraints: $\beta_{kv} > 0$ and $\sum_{v} \beta_{kv} = 1$, meaning that only a handful of vocabulary $v$ contribute to a topic $k$. + +## Let's modify the decoder part + +```{r echo = T} +build.etm.decoder <- + nn_module(classname = "ETM decoder", + initialize = function(n.out, K, jitter = 1e-2) { + self$lbeta <- nn_parameter(torch_randn(K, n.out) * jitter) + self$beta <- nn_log_softmax(2) # topic x variant (softmax for each topic) + self$hid <- nn_log_softmax(2) # sample x topic (softmax for each sample) + }, + ## Define how do get back high-dim data + forward = function(z.b, eps = 1e-8){ + .beta <- self$get.weight() + h.b <- self$hid(z.b) + torch_log(torch_mm(torch_exp(h.b), torch_exp(.beta)) + eps) + }, + ## Helper function + get.weight = function(){ + self$beta(self$lbeta) + }) +``` + +## ETM: putting the encoder and decoder together + +```{r echo = T} +build.etm <- + nn_module( + classname = "embedded topic model", + initialize = function(d.data, K){ + self$enc <- build.vae.encoder(d.data, K) + self$dec <- build.etm.decoder(d.data, K) + }, + forward = function(x){ + .enc <- self$enc(x) + x.hat <- self$dec(.enc$z) + .llik <- multinom.llik(x, x.hat) + .kl <- kl.loss(.enc$z.mean, .enc$z.logvar) + .loss <- .kl - .llik + list(loss = .loss, kl = .kl, latent = .enc$z.mean) + }) +``` + +## ETM: Check flow from the encoder to decoder + +:::::: {.columns} +::: {.column width=.5} + +```{r echo = T} +etm <- build.etm(ncol(x.torch), K=12) +etm$to(device = GPU) + +x.input <- x.torch[1:5, ] +out <- etm(x.input) + +out <- vae(x.input) +out$loss +``` + +::: +::: {.column width=.5} + +```{r echo = T} +############################################# +## reconstruction of x based on the latent ## +############################################# +z.b <- etm$enc(x.input) +logx.recon <- etm$dec(z.b$z) +logx.recon[, 1:5] +``` + +::: +:::::: + +* Note: the reconstructed data matrix is in logarithm scale + +## Results: SGD minimized the loss function + +```{r} +.file <- fig.dir %&% "/etm.rds" +if.needed(.file, { + train.model(x.torch, etm, .file, max.epoch = 2000, record=50) +}) +``` + +```{r} +.result <- readRDS(.file) +``` + +```{r fig.width=5, fig.height = 3} +nepoch <- length(.result$loss) +.df <- data.frame(epoch = seq(0, nepoch - 1), loss = .result$loss) +ggplot(.df, aes(epoch, loss)) + + geom_point(stroke = 0) +``` + +```{r} +show.struct <- function(.zz){ + .hh <- t(apply(t(.zz), 2, function(z) { + pr <- exp(z - max(z)) + pr/sum(pr) + })) + .K <- ncol(.zz) + .dt <- reshape2::melt(.hh) %>% + mutate(row = Var2, col = Var1, weight=value) %>% + col.order(.ro=rev(1:.K), ret.tab=TRUE) %>% + as.data.table() + + .gg.plot(.dt, aes(`col`, `weight`, fill=factor(`Var2`, 1:.K))) + + ylab("topic proportion") + + xlab("cells") + + theme(legend.position = "top") + + theme(axis.text.x = element_blank()) + + theme(axis.ticks.x = element_blank()) + + geom_bar(stat="identity", linewidth=0) + + scale_fill_brewer("latent topic", palette = "Paired") +} +``` + + +## Latent dimensions in the ETM model + +```{r} +tt <- 41 +.df <- as.data.frame(.result$latent[[tt]]) +``` + +```{r fig.width=6, fig.height=2.2, only.plot="1"} +p1 <- ggplot(.df, aes(V1, V2)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p2 <- ggplot(.df, aes(V3, V4)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p3 <- ggplot(.df, aes(V5, V6)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p1 | p2 | p3 +``` + +```{r fig.width=6, fig.height=2.2, only.plot="2"} +p1 <- ggplot(.df, aes(V7, V8)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p2 <- ggplot(.df, aes(V9, V10)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p3 <- ggplot(.df, aes(V11, V12)) + ggrastr::rasterise(geom_point(stroke=0, alpha=.5, size=.5),dpi=300) +p1 | p2 | p3 +``` + +## In ETM, the hidden dimensions are not independent + +\large + +$$H_{ik} = \frac{\exp(Z_{ik})}{\sum_{k'} \exp(Z_{ik'})}$$ + + +## Hidden representations (mixing proportions) + +```{r fig.width=5, fig.height=2.5, only.plot="1"} +tt <- 1 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="2"} +tt <- 2 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="3"} +tt <- 5 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="4"} +tt <- 10 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="5"} +tt <- 20 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="6"} +tt <- 30 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +```{r fig.width=5, fig.height=2.5, only.plot="7"} +tt <- 41 +.temp <- .result$latent[[tt]] +show.struct(.temp) + ggtitle("epoch = " %&% (1 + (tt - 1) * 50)) +``` + +## Annotate factors to cell types by enrichment (`fgsea`) + +```{r run_gsea_etm_beta} +.beta <- exp(t(.result$weights[[tt]])) +.rows <- fread(raw.data$row, header=F) +.rows[, c("gene","chr") := tstrsplit(`V1`, split="[_]+", keep=1:2)] +rownames(.beta) <- .rows$gene + +.db <- read.panglao("Pancreas") +.genes <- unique(unlist(.db)) +.dt <- sort.beta(.beta, genes.selected = .genes) +.gsea <- run.beta.gsea(.dt, .db) +.show <- select.gsea.beta(.gsea, .dt, sd.cutoff = 1e-4) +``` + +```{r} +p1 <- + .gg.plot(.show$beta, aes(row, variable, fill=pmin(`value`, 1e-2))) + + theme(axis.text.x = element_blank()) + + theme(axis.ticks.x = element_blank()) + + theme(legend.position = "none") + + ggrastr::rasterise(geom_tile(), dpi=300) + + ylab("ETM topics") + xlab("genes") + + scale_fill_distiller(direction=1, palette="PuRd", trans="sqrt") + +p2 <- + .gg.plot(.show$leading.edges, aes(row, ct)) + + theme(axis.text.x = element_blank()) + + theme(axis.ticks.x = element_blank()) + + theme(panel.grid = element_blank()) + + geom_tile(linewidth=0) + + ylab("cell types") + +p3 <- + .gg.plot(.show$gsea, aes(variable, ct, fill=-log10(pmax(padj, 1e-4)))) + + xlab("NMF factors") + ylab("") + + scale_y_discrete(position = "right") + + geom_tile(linewidth=.1, color="black") + + theme(legend.key.width=unit(.2, "lines")) + + theme(legend.text = element_text(size=4)) + + scale_fill_distiller("Padj", palette = "YlGnBu", direction=1, labels=function(x) num.sci(10^(-x))) +``` + +```{r fig.width=6, fig.height=3, only.plot="1"} +print(p1/p2) +``` + +```{r fig.width=6, fig.height=3, only.plot="2"} +C <- length(unique(.show$gsea$ct)) +K <- length(unique(.show$beta$variable)) +p0 <- ggplot() + theme_void() +wrap_plots(p1, p0, p2, p3, nrow = 2, heights=c(K,C), widths=c(4,1)) +``` + +## Discussions on latent topic modelling + +* Most cells predominantly belong to one topic (one colour). Why? + +* If we model cells as a mixture of cell topics, we can capture doublets or triplets. + +* The underlying generative model assumes no sequencing depth! This can help avoid batch-specific differences in practice. + +* VAE offers a flexible framework with which our scientific hypothesis can be formulated in a probabilistic language (`torch`). + +* Potentially, this purely-unsupervised learning framework can be combined with supervised, semi-supervised learning models. diff --git a/lect18-scRNA-vae/torch_data.R b/lect18-scRNA-vae/torch_data.R new file mode 100644 index 0000000..47f93c8 --- /dev/null +++ b/lect18-scRNA-vae/torch_data.R @@ -0,0 +1,55 @@ +if(!require(torch)) install.package("torch") +if(!require(mmutilR)) remotes::install_github("causalpathlab/mmutilR") + +scdata.loader <- dataset( + name = "single-cell data loader", + #' @param .hdr single-cell fileset header + #' @param MY.DEV torch device + initialize = function(.hdr, MY.DEV = torch_device("cpu")) { + self$sc.data <- sc.data <- self$.add.ext(.hdr) + self$DEV <- MY.DEV + self$info <- mmutilR::rcpp_mmutil_info(self$sc.data$mtx) + ncells <- self$info$max.col + self$mem.idx <- mmutilR::rcpp_mmutil_read_index(sc.data$idx) + self$read.full() + }, + read.full = function() { + .mtx <- self$sc.data$mtx # matrix market data file + .idx <- self$mem.idx # disk memory index file + .loc <- 1:self$info$max.col # simply read + .out <- mmutilR::rcpp_mmutil_read_columns_sparse(.mtx, .idx, .loc) + .ind <- rbind(.out$col, .out$row) # column is row & vice versa + .sz <- c(.out$max.col, .out$max.row) + xx <- torch_sparse_coo_tensor(.ind, .out$val, .sz, + dtype = torch_float()) + xx <- xx$to_dense() + self$x.list <- lapply(1:nrow(xx), function(r) { + cat(r,"\r",file=stderr());flush(stderr()) + xx[r, , drop=FALSE]$to_sparse()$to(dtype = torch_float16(), + device=self$DEV) + }) + message("Load the full data") + }, + .getitem = function(.loc) { + torch_cat(self$x.list[.loc], dim=1)$to_dense()$to(dtype=torch_float32(), device=self$DEV) + }, + .length = function() { + self$info$max.col + }, + .dim = function() { + self$info$max.row + }, + dim = function() { + self$.dim() + }, + .add.ext = function(.hdr){ + .names <- c("mtx","row","col","idx") + paste0(.hdr, c(".mtx.gz", ".rows.gz", ".cols.gz", ".mtx.gz.index")) %>% + as.list() %>% + (function(x) { names(x) <- .names; x }) + }, + .files.exist = function(...){ + lapply(..., file.exists) %>% + unlist() %>% + all() + }) diff --git a/lect18-scRNA-vae/torch_etm_train.R b/lect18-scRNA-vae/torch_etm_train.R new file mode 100644 index 0000000..9dd9d15 --- /dev/null +++ b/lect18-scRNA-vae/torch_etm_train.R @@ -0,0 +1,121 @@ + +#' @param .mean mean vector +#' @param .lnvar log-variance (independent) +normal.stoch <- function(.mean, .lnvar) { + .eps <- torch_randn_like(.lnvar) + .sig <- torch_exp(.lnvar / 2.) + .mean + .eps * .sig +} + +kl.loss <- function(.mean, .lnvar) { + -0.5 * torch_sum(1. + .lnvar - torch_pow(.mean, 2.) - torch_exp(.lnvar), dim = -1); +} + +#' FC ReLU layer generator +#' @param in_ input dimension +#' @param out_ output dimension +#' @return a generator +#' +build.fc.relu <- function(in_, out_) { + nn_sequential(nn_linear(in_, out_), + nn_relu()) +} + +#' Build a stack of layers +#' @param in_ input dimension (first layer) +#' @param layers dimensions for the subsequent layers +#' @param generator a shared generator function +#' @param name a header for the names +#' +build.stack <- function(in_, layers, generator = nn_linear, name = "", stdizer = NULL) { + ret <- + nn_module( + classname = "stack.layers", + initialize = function(in_, + .layers, + .generator, + .name) { + d.prev <- in_ + for(l in 1:length(.layers)) { + d.this <- .layers[l] + .name.l <- paste0(name, ".a", (l - 1)) + self$add_module(.name.l, module = .generator(d.prev, d.this)) + if(!is.null(stdizer)){ + .name.l <- paste0(name, ".d", (l - 1)) + self$add_module(.name.l, module = stdizer(d.this)) + } + d.prev <- d.this + } + }, + forward = function(input) { + for(module in private$modules_) { + input <- module(input) + } + input + }) + return(ret(in_, layers, generator, name)) +} + +#' @param n.in +#' @param K +#' @param d +build.etm.encoder <- + nn_module( + classname = "Encoder", + initialize = function(n.in, K, layers) { + self$fc <- build.stack(n.in, layers, + generator = build.fc.relu, + name = "enc.fc") + d <- layers[length(layers)] + self$K <- K + self$z.mean <- nn_linear(d, K) + self$z.lnvar <- nn_linear(d, K) + }, + forward = function(xx) { + xx <- nnf_normalize(xx, dim=2) + ss <- self$fc(xx) + mm <- self$z.mean(ss) + lv <- torch_clamp(self$z.lnvar(ss), -4.0, 4.0) + z <- normal.stoch(mm, lv) + list(z = z, z.mean = mm, z.lnvar = lv) + }) + +#' @param n.out +#' @param K +#' @param jitter +build.etm.decoder <- + nn_module( + classname = "Decoder", + initialize = function(n.out, K, jitter = .1) { + self$lbeta <- nn_parameter(torch_randn(K, n.out) * jitter) + self$beta <- nn_log_softmax(2) # topic x variant (softmax for each topic) + self$hid <- nn_log_softmax(2) # sample x topic (softmax for each sample) + }, + forward = function(zz) { + .beta <- self$beta(self$lbeta) + .hh <- self$hid(zz) + torch_mm(torch_exp(.hh), torch_exp(.beta)) + }) + +#' @param n.in input data dimension +#' @param n.out output data dimension +#' @param K latent state dimension +build.etm <- + nn_module( + classname = "ETM", + initialize = function(n.in, n.out, K, d = 16) { # Register model parametesr + self$enc <- build.etm.encoder(n.in, K, layers=d) # encoder model + self$dec <- build.etm.decoder(n.out, K) # decoder model + }, + forward = function(xx) { # Define the forward pass + .enc <- self$enc(xx) # Converts data to the latent + .pr <- self$dec(.enc$z) # Converts the latent to recon + list(recon = .pr, z.mean = .enc$z.mean, z.lnvar = .enc$z.lnvar) + }) + + +#' @param xx observed data (vocabulary frequency) +#' @param pr reconstructed probability of observing vocabulary +etm.llik <- function(xx, pr, eps = 1e-8){ + torch_sum(xx * torch_log(pr + eps), dim=-1) +}

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