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JavaScriptで特異値分解(SVD)したい

2021/12/04に公開約10,100字

ちょっとした解析や機械学習をしようとすると、けっこうな頻度で行列の固有値を求めたくなります。
そして行列が大抵の場合正方行列でなく、特異値分解(SVD)が必要になります。

だいたいの解説では「ライブラリを使って〜」の一行で片付けられてしまうところなのですが、自分でもJavaScript(TypeScript)で書いてみようという車輪の再発明記事です。

G. H. Golub, C. ReinschによるSingular Value Decomposition and Least Squares Solutionsのアルゴリズムの写経になります。
論文を忠実になぞっているだけなので、元論文に経緯さえ払えばコピペはご自由に。
とりあえず解ければいい人向けで、より高速だったり精度がいい解法を求める人は自前実装など捨ててLAPACKなどを使いましょう。

const svd = (A: number[][], options?: {
    eps?: number;
    beta?: number;
    maxIter?: number;
}): [number[], number[][], number[][]] => {
    const maxIter = options?.maxIter ? options.maxIter : 50;
    const beta = options?.beta ? options.beta : Number.MIN_VALUE;
    // beta is the smallest positive number representable in the computer.
    // https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Number/MIN_VALUE
    let eps = options?.eps ? options.eps : Number.EPSILON; // the machine precision

    const tol = beta / eps;

    let m = A.length;
    if (m < 1) {
        throw new Error("row size must be more than 0.");
    }
    let n = A[0].length;
    if (n < 1) {
        throw new Error("col size a must be more than 0.");
    }

    // if m < n, the algorithms may be applied to At
    // At = (U Q Vt)t = V Q Ut
    const portrait = m >= n;
    const U = portrait ? copy(A) : copy(transpose(A));
    if (!portrait) {
        [m, n] = [n, m];
    }

    const q = vZeros(n);
    const e = vZeros(n);
    const V = mZeros(n);

    // Householeder's reduction to bidiagonal form
    let g = 0, x = 0;
    for (let i = 0; i < n; i++) {
        e[i] = g;
        let s = 0;
        const l = i + 1;
        for (let j = i; j < m; j++) {
            s += U[j][i] ** 2;
        }
        if (s < tol) {
            g = 0;
        } else {
            const f = U[i][i];
            g = f < 0 ? Math.sqrt(s) : -Math.sqrt(s);
            const h = f * g - s;
            U[i][i] = f - g;
            for (let j = l; j < n; j++) {
                let s = 0;
                for (let k = i; k < m; k++) {
                    s += U[k][i] * U[k][j];
                }
                const f = s / h;
                for (let k = i; k < m; k++) {
                    U[k][j] += f * U[k][i];
                }
            }
        }
        q[i] = g;
        s = 0;
        for (let j = l; j < n; j++) {
            s += U[i][j] ** 2;
        }
        if (s < tol) { // When l >= n, s is always 0.
            g = 0;
        } else {
            const f = U[i][l];
            g = f < 0 ? Math.sqrt(s) : -Math.sqrt(s);
            const h = f * g - s;
            U[i][l] = f - g;
            for (let j = l; j < n; j++) {
                e[j] = U[i][j] / h;
            }
            for (let j = l; j < m; j++) {
                let s = 0;
                for (let k = l; k < n; k++) {
                    s += U[j][k] * U[i][k];
                }
                for (let k = l; k < n; k++) {
                    U[j][k] += s * e[k];
                }
            }
        }
        const y = Math.abs(q[i]) + Math.abs(e[i]);
        if (y > x) x = y;
    }
    // accumulation of right-hand transformations
    for (let i = n - 1; i >= 0; i--) {
        const l = i + 1;
        if (g !== 0) { // When i = n - 1, g is always 0.
            const h = U[i][l] * g;
            for (let j = l; j < n; j++) {
                V[j][i] = U[i][j] / h;
            }
            for (let j = l; j < n; j++) {
                let s = 0;
                for (let k = l; k < n; k++) {
                    s += U[i][k] * V[k][j];
                }
                for (let k = l; k < n; k++) {
                    V[k][j] += s * V[k][i];
                }
            }
        }
        for (let j = l; j < n; j++) {
            V[i][j] = 0;
            V[j][i] = 0;
        }
        V[i][i] = 1;
        g = e[i];
    }
    // accumulation of left-hand transformations
    for (let i = n - 1; i >= 0; i--) {
        const l = i + 1;
        g = q[i];
        for (let j = l; j < n; j++) {
            U[i][j] = 0;
        }
        if (g !== 0) {
            const h = U[i][i] * g;
            for (let j = l; j < n; j++) {
                let s = 0;
                for (let k = l; k < m; k++) {
                    s += U[k][i] * U[k][j];
                }
                const f = s / h;
                for (let k = i; k < m; k++) {
                    U[k][j] += f * U[k][i];
                }
            }

            for (let j = i; j < m; j++) {
                U[j][i] /= g;
            }
        } else {
            for (let j = i; j < m; j++) {
                U[j][i] = 0;
            }
        }
        U[i][i] += 1;
    }

    // diagonalization of bidiagonal form
    eps *= x;
    for (let k = n - 1; k >= 0; k--) {
        let z = Infinity;
        let cnt = 0
        while (++cnt) {
            // test f splitting
            let l, convergence = false;
            for (l = k; l >= 0; l--) {
                if (Math.abs(e[l]) <= eps) { // e[0] is always 0.
                    convergence = true;
                    break;
                }
                if (Math.abs(q[l - 1]) <= eps) {
                    break;
                }
            }
            // cancellation of e[l]
            if (!convergence) {
                let c = 0, s = 1;
                const l1 = l - 1; // alyways l1 >= 0
                for (let i = l; i <= k; i++) {
                    const f = s * e[i];
                    e[i] *= c;
                    if (Math.abs(f) <= eps)
                        break;
                    const g = q[i];
                    const h = Math.sqrt(f ** 2 + g ** 2);
                    q[i] = h;
                    c = g / h;
                    s = -f / h;

                    for (let j = 0; j < m; j++) {
                        const y = U[j][l1];
                        const z = U[j][i];
                        U[j][l1] = y * c + z * s;
                        U[j][i] = -y * s + z * c;
                    }
                }
            }
            // test f convergence
            z = q[k];
            if (l === k) break; // when k = 0, l is always 0.
            if (cnt >= maxIter) {
                throw new Error();
            }

            // shift from bottom 2*2 minor
            let x = q[l], g = e[k - 1];
            const y = q[k - 1], h = e[k];
            let f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2 * h * y);
            g = Math.sqrt(f ** 2 + 1);
            f = ((x - z) * (x + z) + h * (y / (f < 0 ? (f - g) : (f + g)) - h)) / x;

            // next QR transformation
            let c = 1, s = 1;
            for (let i = l + 1; i <= k; i++) { // k <= n - 1, l >= 0
                let g = e[i], y = q[i], h = s * g;
                g *= c;
                let z = Math.sqrt(f ** 2 + h ** 2);
                e[i - 1] = z;
                c = f / z;
                s = h / z;
                f = x * c + g * s;
                g = -x * s + g * c;
                h = y * s;
                y *= c;
                for (let j = 0; j < n; j++) {
                    const x = V[j][i - 1];
                    const z = V[j][i];
                    V[j][i - 1] = x * c + z * s;
                    V[j][i] = -x * s + z * c;
                }
                z = Math.sqrt(f ** 2 + h ** 2);
                q[i - 1] = z;
                c = f / z;
                s = h / z;
                f = c * g + s * y;
                x = -s * g + c * y;
                for (let j = 0; j < m; j++) {
                    const y = U[j][i - 1];
                    const z = U[j][i];
                    U[j][i - 1] = y * c + z * s;
                    U[j][i] = -y * s + z * c;
                }
            }
            e[l] = 0;
            e[k] = f;
            q[k] = x;
        }

        if (z < 0) {
            // q[k] is made non-negative
            q[k] = -z;
            for (let j = 0; j < n; j++) {
                V[j][k] *= -1;
            }
        }
    }
    return portrait ? [q, U, V] : [q, V, U];
}

const vZeros = (m: number): number[] => {
    if (m < 1) {
        throw new Error("m must be more than 0.");
    }
    return (Array(m) as number[]).fill(0);
}

const mZeros = (row: number, col?: number): number[][] => {
    if (row < 1) {
        throw new Error("row must be more than 0.");
    }
    if (col && col < 1) {
        throw new Error("col must be more than 0.");
    }
    const m = row;
    const n = col ? col : row;

    const ret: number[][] = [];
    for (let i = 0; i < m; i++) {
        ret.push(vZeros(n));
    }
    return ret;
}

const copy = (mat: number[][]): number[][] => {
    const m = mat.length;
    const ret = [];
    for (let i = 0; i < m; i++) {
        const row = [];
        const n = mat[i].length;
        for (let j = 0; j < n; j++) {
            row.push(mat[i][j]);
        }
        ret.push(row);
    }
    return ret;
}

const transpose = (mat: number[][]): number[][] => {
    const ret: number[][] = [];
    const m = mat.length;
    const n = mat[0]?.length;

    for (let i = 0; i < n; i++) {
        const row = [];
        for (let j = 0; j < m; j++) {
            row.push(mat[j][i]);
        }
        ret.push(row);
    }
    return ret;
}

const mult = (mat1: number[][], mat2: number[][]): number[][] => {
    if (mat1[0]?.length !== mat2.length) {
        throw new Error("Matrices don't match size.");
    }
    const ret: number[][] = [];

    for (let i = 0; i < mat1.length; i++) {
        const row = [];
        for (let j = 0; j < mat2[0].length; j++) {
            let elm = 0;
            for (let k = 0; k < mat1[0].length; k++) {
                elm += mat1[i][k] * mat2[k][j];
            }
            row.push(elm)
        }
        ret.push(row);
    }
    return ret;
}

唯一手を加えている点は、このアルゴリズムは縦長の行列(m>=n)にしか対応していませんが、転置して計算してからUとVを入れ替えれば結果は同じなので、そこだけ処理を追加しています。

最近の端末はみんなスペックが高いので、よほど大きな行列でもない限りはサクサク計算してくれます。

車輪の再発明を終えてから気が付きましたが、数は少ないが世に出回っているJavaScript製SVDライブラリたちもほとんどは同じ論文を元に作られていますね。アルゴリズムの記述がシンプルでわかりやすく実装しやすいからでしょう。ありがたい論文です。

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