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* Pearson's *r* between every pair of rows of a square, over the cells both rows
* have — the correlation matrix of §18.6, fast enough to draw at 4,096 bins.
*
* It was computed pair by pair with `stats.ts`'s `pearson`: two passes over two
* rows for each of `m²/2` pairs, a `NaN` test on every cell of both, in scalar
* JavaScript. That is right and slow — 2.8 s at 929 bins, and three minutes at
* 4,096, where the map's correlation was capped at 1,500 because nobody should be
* asked to wait longer for a toggle. The arithmetic is the same here and is
* organised differently.
*
* **One product, and a few numbers per pair.** Over the cells a pair shares, `r`
* needs six sums: the pair's products, how many cells there are, and each row's
* sum and sum of squares over them. Only the first depends on both rows at every
* cell; the other five are each row's own totals, less what the *other* row is
* missing. So every row is read once, a missing cell is written as zero, and
*
* - the products of every pair are one matrix product — the rows against
* themselves, `m³/2` multiply-adds — done by a WebAssembly kernel with SIMD
* (`wasm/gram.c`) at ten to thirteen billion a second on one core, the more
* where the engine fuses the multiply-add, where the pairwise loop managed a
* fifth of a billion;
* - the other five are each row's totals, corrected pair by pair for the few
* cells the other row lacks — a handful on a balanced map, where a bin the
* balancer dropped is dropped from both sides of the square and what is left
* missing is the odd cell with no expectation, or the diagonals the
* compartment chain ignores.
*
* A square where rows miss whole stretches — a trans-only compartment run blanks
* every sequence's own block — would make those corrections as costly as the
* product, so there the kernel sums all six over the shared cells in one masked
* pass instead (`gram_masked`), about seven times the work of the plain product
* and still far below the pairwise loop. What the corrections cost between the
* two is real: a square with a few hundred cells missing from every row — the
* map's O/E of a sparse map did, when its empty pixels were blank — is several
* times the product either way, which is why the map correlates a ratio whose
* empty pixels are zeros (`render/derive.ts`'s `keepZeros`).
*
* The work comes in **bands** of rows, each the band's rows against every row
* after them, so a band can be handed to another thread with only the rows it
* reads (`pearsonThreads.ts`), and a caller hears about every row as its band
* finishes.
*
* **Precision.** `stats.ts` computes `r` in two passes because the one-pass form,
* built from `Σx`, `Σx²` and `Σxy`, loses its significant digits when the mean is
* large next to the spread — which is an observed/expected row, centred on one.
* Each row here is shifted by its own mean before anything is summed, and `r` is
* blind to such a shift, so the sums are of numbers centred on zero and nothing
* large cancels: the pair's own mean differs from the row's only through the few
* cells the other row is missing. The rows are held as `float` and each block of
* 512 cells is summed in `float` lanes before it reaches a `double`, so `r` agrees
* with the two-pass `double` computation to about 1e-6 — below anything a
* colour ramp or an eigenvector can show.
*
* **What it gives back** is the same as before: `NaN` for a pair with fewer than
* {@link MIN_PAIRS} cells in common or a row with no spread over them, which the
* callers read as *no evidence* — and each row's line is handed over as soon as
* it is done, so neither caller holds a second copy of the square.
*/
import { MIN_PAIRS } from './stats'
import { GRAM_RELAXED_WASM_BASE64, GRAM_WASM_BASE64 } from './wasm/gram.generated'
/**
* The cells summed per kernel call along a row. 512 floats is 2 KB a row, so the
* rows being multiplied stay in the first-level cache while the band passes over
* them; measured at 2,048 bins, 256 and 1,024 are both slower.
*/
const K_BLOCK = 512
/** The rows multiplied against a band per kernel call: 256 × 2 KB, a second-level cache's worth. */
const B_BLOCK = 256
/**
* The rows of the output computed per pass, and so the step progress is reported
* and a cancel is heard at: the first band of a 4,096-bin square is about a
* tenth of a second on one core.
*/
export const BAND = 64
/**
* A row whose spread over the shared cells is below this fraction of its sum of
* squares has none: the exact computation would divide by a zero that rounding
* has made a very small number.
*/
const FLAT = 1e-9
/** The square, a row at a time. */
export interface PearsonInput {
/** Rows, and columns: the square is `m × m`. */
m: number
/** Row `r`'s `m` cells into `into`, anything but a finite number where it has none. */
row(r: number, into: Float64Array): void
}
/**
* Row `a`'s correlations with every later row, in `line[a + 1 .. m)`, handed over
* once per row and in order. `NaN` where a pair has no correlation.
*/
export type PearsonOutput = (a: number, line: Float64Array) => void
/**
* Which arithmetic runs the product: the fastest WebAssembly kernel this engine
* runs, the one with SIMD alone — what an engine without relaxed SIMD gets, and
* what the tests hold beside it — or the same loops in JavaScript.
*/
export type PearsonEngine = 'wasm' | 'simd' | 'js'
/**
* Rows `[first, m)` of a square, read and ready to be correlated — everything a
* band needs, and so everything a thread computing bands is sent
* (`pearsonThreads.ts`).
*/
export interface PreparedRows {
/** The square's side. */
m: number
/** Floats between one row and the next: `m` rounded up to a multiple of four. */
ld: number
/** The first row held; every array below starts at it. */
first: number
/**
* Whether the rows are in the masked kernel's form — `NaN` where a cell is
* missing — because too many are for pair-by-pair corrections to be cheaper.
* Otherwise a missing cell is zero, and `starts`/`cols` list them.
*/
masked: boolean
/** Each row's finite cells, their sum and their sum of squares — of the shifted, stored values. */
count: Float64Array
sum: Float64Array
squares: Float64Array
/** Row `first + i`'s missing columns are `cols[starts[i] .. starts[i + 1])`, sorted; empty when masked. */
starts: Int32Array
cols: Int32Array
}
/**
* Correlate every pair of rows, on this thread.
*
* `onRow` is called for every row, in order, before its correlations are
* computed, and **awaited** — which is what lets a caller report progress and
* stop it by throwing (`analysis/job.ts`'s `breathe` explains why the yield has
* to be inside the loop).
*/
export async function pearsonRows(
input: PearsonInput,
output: PearsonOutput,
onRow?: (row: number) => void | Promise<void>,
options: { engine?: PearsonEngine } = {},
): Promise<void> {
const m = input.m
if (m <= 0) return
const { engine, rows } = readSquare(input, options.engine ?? 'wasm')
const line = new Float64Array(m)
for (let r0 = 0; r0 < m; r0 += BAND) {
const r1 = Math.min(m, r0 + BAND)
if (onRow !== undefined) {
for (let r = r0; r < r1; r++) {
const waiting = onRow(r)
if (waiting !== undefined) await waiting
}
}
correlateBand(engine, rows, r0, r1, line, output)
}
}
/** The floats between one row and the next: `m` rounded up to whole vectors of four. */
export function strideOf(m: number): number {
return (m + 3) & ~3
}
/**
* Read the whole square into an engine's storage, in the form the product it
* needs wants: the first step of every road, threaded or not.
*/
export function readSquare(
input: PearsonInput,
wanted: PearsonEngine = 'wasm',
): { engine: RowsEngine; rows: PreparedRows } {
const m = input.m
const ld = strideOf(m)
const engine = makeEngine(wanted, ld, m)
const x = engine.x
const count = new Float64Array(m)
const sum = new Float64Array(m)
const squares = new Float64Array(m)
// Past an eighth of the square missing, the corrections cost more than the
// masked pass: each is a gather per missing cell per pair, against seven
// multiply-adds per cell per pair in the kernel, which runs about eight times
// faster than a gather.
const limit = Math.floor((m * m) / 8)
const starts = new Int32Array(m + 1)
let cols = new Int32Array(Math.min(limit, Math.max(64, m)))
let listed = 0
let masked = false
const scratch = new Float64Array(m)
for (let r = 0; r < m; r++) {
input.row(r, scratch)
let total = 0
let seen = 0
for (let c = 0; c < m; c++) {
const value = scratch[c]
if (Number.isFinite(value)) {
total += value
seen++
}
}
const mean = seen > 0 ? total / seen : 0
starts[r] = listed
const base = r * ld
let s = 0
let q = 0
for (let c = 0; c < m; c++) {
const value = scratch[c]
if (Number.isFinite(value)) {
const shifted = Math.fround(value - mean)
x[base + c] = shifted
s += shifted
q += shifted * shifted
continue
}
// `NaN` for now, which is the masked kernel's form; the plain one's is
// made from it below, once it is known which is wanted.
x[base + c] = Number.NaN
if (masked) continue
if (listed === limit) {
masked = true
continue
}
if (listed === cols.length) {
const grown = new Int32Array(Math.min(limit, cols.length * 2))
grown.set(cols)
cols = grown
}
cols[listed++] = c
}
for (let c = m; c < ld; c++) x[base + c] = Number.NaN
count[r] = seen
sum[r] = s
squares[r] = q
}
starts[m] = listed
if (!masked) {
// The plain kernel wants a missing cell, and the padding, as zero.
for (let r = 0; r < m; r++) {
for (let at = starts[r]; at < starts[r + 1]; at++) x[r * ld + cols[at]] = 0
for (let c = m; c < ld; c++) x[r * ld + c] = 0
}
}
return {
engine,
rows: {
m,
ld,
first: 0,
masked,
count,
sum,
squares,
starts: masked ? new Int32Array(m + 1) : starts,
cols: masked ? new Int32Array(0) : cols.subarray(0, listed),
},
}
}
/**
* The correlations of rows `[r0, r1)` with every row after each, handed to
* `output` a row at a time through `line`. The engine holds rows from
* `rows.first` on, which must be no later than `r0`.
*/
export function correlateBand(
engine: RowsEngine,
rows: PreparedRows,
r0: number,
r1: number,
line: Float64Array,
output: PearsonOutput,
): void {
const { m, first } = rows
const held = m - first
engine.run(r0 - first, r1 - first, rows.masked)
const out = engine.out
for (let a = r0; a < r1; a++) {
const g = (a - r0) * held - first
if (rows.masked) {
const size = BAND * held
for (let b = a + 1; b < m; b++) {
const at = g + b
line[b] = correlation(
out[at],
out[size + at],
out[2 * size + at],
out[3 * size + at],
out[4 * size + at],
out[5 * size + at],
)
}
} else {
sparseLine(a, engine.x, rows, out, g, line)
}
output(a, line)
}
}
/**
* Row `a`'s correlations from the product band and the corrections: each row's
* totals, less its cells at the columns the other row is missing.
*/
function sparseLine(a: number, x: Float32Array, rows: PreparedRows, band: Float64Array, g: number, line: Float64Array): void {
const { m, ld, first, count, sum, squares, starts, cols } = rows
const ia = a - first
const aFrom = starts[ia]
const aTo = starts[ia + 1]
const aBase = ia * ld
for (let b = a + 1; b < m; b++) {
const ib = b - first
let n = count[ia]
let sa = sum[ia]
let qa = squares[ia]
let sb = sum[ib]
let qb = squares[ib]
const bFrom = starts[ib]
const bTo = starts[ib + 1]
if (bTo > bFrom) {
// Row a's cells at b's gaps come out of a's totals. Where a is missing
// too, the stored zero takes nothing out, and the merge below counts it.
for (let at = bFrom; at < bTo; at++) {
const value = x[aBase + cols[at]]
sa -= value
qa -= value * value
}
n -= bTo - bFrom
}
if (aTo > aFrom) {
const bBase = ib * ld
for (let at = aFrom; at < aTo; at++) {
const value = x[bBase + cols[at]]
sb -= value
qb -= value * value
}
// Columns both are missing were taken off a's count once above as b's
// gaps; they were never among a's cells, so they are given back.
if (bTo > bFrom) n += shared(cols, aFrom, aTo, bFrom, bTo)
}
line[b] = correlation(band[g + b], n, sa, sb, qa, qb)
}
}
/** How many columns two sorted runs of `cols` have in common. */
function shared(cols: Int32Array, i0: number, i1: number, j0: number, j1: number): number {
let both = 0
let i = i0
let j = j0
while (i < i1 && j < j1) {
const u = cols[i]
const v = cols[j]
if (u === v) {
both++
i++
j++
} else if (u < v) i++
else j++
}
return both
}
/** `r` from the six sums over the shared cells, or `NaN` where it has none. */
function correlation(p: number, n: number, sa: number, sb: number, qa: number, qb: number): number {
if (!(n >= MIN_PAIRS)) return Number.NaN
const va = qa - (sa * sa) / n
const vb = qb - (sb * sb) / n
if (!(va > FLAT * qa) || !(vb > FLAT * qb)) return Number.NaN
const r = (p - (sa * sb) / n) / Math.sqrt(va * vb)
return r > 1 ? 1 : r < -1 ? -1 : r
}
// ---------------------------------------------------------------------------
// The product
// ---------------------------------------------------------------------------
/** Where the rows are held, and the arithmetic over them. */
export interface RowsEngine {
/** The rows held, `ld` floats apart, counted from the first held. */
readonly x: Float32Array
/** The band's sums: one `BAND × held` block for the plain product, six for the masked one. */
readonly out: Float64Array
/** Fill the band for held rows `[r0, r1)` against every held row from `r0` on. */
run(r0: number, r1: number, masked: boolean): void
}
type Kernel = (
x: number,
y: number,
ld: number,
out: number,
ldg: number,
g0: number,
a0: number,
a1: number,
b0: number,
b1: number,
k0: number,
k1: number,
) => void
type MaskedKernel = (
x: number,
ld: number,
out: number,
ldg: number,
size: number,
g0: number,
a0: number,
a1: number,
b0: number,
b1: number,
k0: number,
k1: number,
) => void
/**
* The kernel's two builds, compiled once per thread; each `null` where this
* engine cannot run it — relaxed SIMD is newer than SIMD, and an engine with
* neither has no kernel at all.
*/
let compiled: { relaxed: WebAssembly.Module | null; simd: WebAssembly.Module | null } | undefined
function compile(base64: string): WebAssembly.Module | null {
try {
if (typeof WebAssembly === 'undefined') return null
const text = atob(base64)
const bytes = new Uint8Array(text.length)
for (let i = 0; i < text.length; i++) bytes[i] = text.charCodeAt(i)
// `validate` is false on an engine without the instructions a build uses,
// which is the one way either can fail to load where WebAssembly exists.
return WebAssembly.validate(bytes) ? new WebAssembly.Module(bytes) : null
} catch {
return null
}
}
/** The build `wanted` names: the relaxed one where it runs, for `wasm`. */
function gramModule(wanted: 'wasm' | 'simd'): WebAssembly.Module | null {
compiled ??= { relaxed: compile(GRAM_RELAXED_WASM_BASE64), simd: compile(GRAM_WASM_BASE64) }
return wanted === 'wasm' ? (compiled.relaxed ?? compiled.simd) : compiled.simd
}
/** Whether the WebAssembly kernel can run here, and which build — for the notes and the tests. */
export function gramKernelAvailable(build: 'wasm' | 'simd' | 'relaxed' = 'wasm'): boolean {
if (build === 'relaxed') {
gramModule('wasm')
return compiled?.relaxed != null
}
return gramModule(build) !== null
}
/**
* Storage for `held` rows, `ld` floats each, and the arithmetic over them.
* Sized for the masked product's six output bands whichever runs — they are a
* few megabytes, and which product a square needs is only known once it has
* been read into this storage.
*/
export function makeEngine(wanted: PearsonEngine, ld: number, held: number): RowsEngine {
const module = wanted === 'js' ? null : gramModule(wanted)
if (module !== null) {
try {
return wasmEngine(module, ld, held)
} catch {
// Memory the size of the square could not be had in one piece: the same
// arithmetic in JavaScript needs none.
}
}
return jsEngine(ld, held)
}
const PAGE = 65536
function wasmEngine(module: WebAssembly.Module, ld: number, held: number): RowsEngine {
const xBytes = held * ld * 4
const outAt = Math.ceil(xBytes / 8) * 8
const outLength = 6 * BAND * held
const memory = new WebAssembly.Memory({ initial: Math.ceil((outAt + outLength * 8) / PAGE) + 1 })
const instance = new WebAssembly.Instance(module, { env: { memory } })
const gram = instance.exports.gram as Kernel
const masked = instance.exports.gram_masked as MaskedKernel
const x = new Float32Array(memory.buffer, 0, held * ld)
const out = new Float64Array(memory.buffer, outAt, outLength)
return {
x,
out,
run(r0, r1, isMasked) {
out.fill(0, 0, isMasked ? out.length : BAND * held)
for (let k0 = 0; k0 < ld; k0 += K_BLOCK) {
const k1 = Math.min(ld, k0 + K_BLOCK)
for (let b0 = r0; b0 < held; b0 += B_BLOCK) {
const b1 = Math.min(held, b0 + B_BLOCK)
if (isMasked) masked(0, ld, outAt, held, BAND * held, r0, r0, r1, b0, b1, k0, k1)
else gram(0, 0, ld, outAt, held, r0, r0, r1, b0, b1, k0, k1)
}
}
},
}
}
function jsEngine(ld: number, held: number): RowsEngine {
const x = new Float32Array(held * ld)
const out = new Float64Array(6 * BAND * held)
return {
x,
out,
run(r0, r1, isMasked) {
out.fill(0, 0, isMasked ? out.length : BAND * held)
for (let k0 = 0; k0 < ld; k0 += K_BLOCK) {
const k1 = Math.min(ld, k0 + K_BLOCK)
for (let b0 = r0; b0 < held; b0 += B_BLOCK) {
const b1 = Math.min(held, b0 + B_BLOCK)
if (isMasked) gramMaskedJs(x, ld, out, held, BAND * held, r0, r0, r1, b0, b1, k0, k1)
else gramJs(x, ld, out, held, r0, r0, r1, b0, b1, k0, k1)
}
}
},
}
}
/**
* `gram.c`'s `gram`, in JavaScript: the fallback where the module cannot load,
* and the reference the tests hold the kernel to. Two rows against two, so the
* four sums stay in registers and each load feeds two of them.
*/
function gramJs(
x: Float32Array,
ld: number,
out: Float64Array,
ldg: number,
g0: number,
a0: number,
a1: number,
b0: number,
b1: number,
k0: number,
k1: number,
): void {
for (let a = a0; a < a1; a += 2) {
const ar = a + 1 < a1 ? a + 1 : a
const pa0 = a * ld
const pa1 = ar * ld
const o0 = (a - g0) * ldg
const o1 = (ar - g0) * ldg
for (let b = Math.max(a, b0); b < b1; b += 2) {
const br = b + 1 < b1 ? b + 1 : b
const pb0 = b * ld
const pb1 = br * ld
let s00 = 0
let s01 = 0
let s10 = 0
let s11 = 0
for (let k = k0; k < k1; k++) {
const u0 = x[pa0 + k]
const u1 = x[pa1 + k]
const v0 = x[pb0 + k]
const v1 = x[pb1 + k]
s00 += u0 * v0
s01 += u0 * v1
s10 += u1 * v0
s11 += u1 * v1
}
out[o0 + b] += s00
if (br !== b) out[o0 + br] += s01
if (ar !== a) {
out[o1 + b] += s10
if (br !== b) out[o1 + br] += s11
}
}
}
}
/** `gram.c`'s `gram_masked`, in JavaScript, for the same two reasons. */
function gramMaskedJs(
x: Float32Array,
ld: number,
out: Float64Array,
ldg: number,
size: number,
g0: number,
a0: number,
a1: number,
b0: number,
b1: number,
k0: number,
k1: number,
): void {
for (let a = a0; a < a1; a++) {
const pa = a * ld
const o = (a - g0) * ldg
for (let b = Math.max(a, b0); b < b1; b++) {
const pb = b * ld
let p = 0
let n = 0
let sa = 0
let sb = 0
let qa = 0
let qb = 0
for (let k = k0; k < k1; k++) {
const u = x[pa + k]
const v = x[pb + k]
if (u !== u || v !== v) continue
p += u * v
n++
sa += u
sb += v
qa += u * u
qb += v * v
}
out[o + b] += p
out[size + o + b] += n
out[2 * size + o + b] += sa
out[3 * size + o + b] += sb
out[4 * size + o + b] += qa
out[5 * size + o + b] += qb
}
}
}
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