Data Analytics: MTTKRP

Matricized tensor times Khatri-Rao product (MTTKRP) is a bottleneck operation in various algorithms - such as Alternating Least Squares - for computing sparse tensor factorizations like the Canonical Polyadic Decomposition. Mathematically, mode-1 MTTKRP (for three-dimensional tensors) can be expressed as

where , , and are typically dense matrices, is a three-dimensional tensor (matricizied along the first mode), and denotes the Khatri-Rao product. This operation can also be expressed in index notation as

You can use the TACO C++ API to easily and efficiently compute the MTTKRP, as shown here:

// On Linux and MacOS, you can compile and run this program like so:
//   g++ -std=c++11 -O3 -DNDEBUG -DTACO -I ../../include -L../../build/lib mttkrp.cpp -o mttkrp -ltaco
//   LD_LIBRARY_PATH=../../build/lib ./mttkrp
#include <random>
#include "taco.h"
using namespace taco;
int main(int argc, char* argv[]) {
  std::default_random_engine gen(0);
  std::uniform_real_distribution<double> unif(0.0, 1.0);
  // Predeclare the storage formats that the inputs and output will be stored as.
  // To define a format, you must specify whether each dimension is dense or 
  // sparse and (optionally) the order in which dimensions should be stored. The 
  // formats declared below correspond to compressed sparse fiber (csf) and 
  // row-major dense (rm).
  Format csf({Sparse,Sparse,Sparse});
  Format  rm({Dense,Dense});

  // Load a sparse order-3 tensor from file (stored in the FROSTT format) and 
  // store it as a compressed sparse fiber tensor. The tensor in this example 
  // can be download from:
  Tensor<double> B = read("nell-2.tns", csf);
  // Generate a random dense matrix and store it in row-major (dense) format. 
  // Matrices correspond to order-2 tensors in taco.
  Tensor<double> C({B.getDimension(1), 25}, rm);
  for (int i = 0; i < C.getDimension(0); ++i) {
    for (int j = 0; j < C.getDimension(1); ++j) {
      C.insert({i,j}, unif(gen));

  // Generate another random dense matrix and store it in row-major format.
  Tensor<double> D({B.getDimension(2), 25}, rm);
  for (int i = 0; i < D.getDimension(0); ++i) {
    for (int j = 0; j < D.getDimension(1); ++j) {
      D.insert({i,j}, unif(gen));

  // Declare the output matrix to be a dense matrix with 25 columns and the same
  // number of rows as the number of slices along the first dimension of input
  // tensor B, to be also stored as a row-major dense matrix.
  Tensor<double> A({B.getDimension(0), 25}, rm);

  // Define the MTTKRP computation using index notation.
  IndexVar i, j, k, l;
  A(i,j) = B(i,k,l) * D(l,j) * C(k,j);
  // At this point, we have defined how entries in the output matrix should be
  // computed from entries in the input tensor and matrices but have not actually
  // performed the computation yet. To do so, we must first tell taco to generate
  // code that can be executed to compute the MTTKRP operation.
  // We can now call the functions taco generated to assemble the indices of the
  // output matrix and then actually compute the MTTKRP.
  // Write the output of the computation to file (stored in the FROSTT format).
  write("A.tns", A);

You can also use the TACO Python API to perform the same computation, as demonstrated here:

import pytaco as pt
import numpy as np
from pytaco import compressed, dense

# Define formats for storing the sparse tensor and dense matrices
csf = pt.format([compressed, compressed, compressed])
rm  = pt.format([dense, dense])

# Load a sparse three-dimensional tensor from file (stored in the FROSTT
# format) and store it as a compressed sparse fiber tensor. The tensor in this
# example can be download from:
B ="nell-2.tns", csf);

# Generate two random matrices using NumPy and pass them into TACO
C = pt.from_array(np.random.uniform(size=(B.shape[1], 25)))
D = pt.from_array(np.random.uniform(size=(B.shape[2], 25)))

# Declare the result to be a dense matrix
A = pt.tensor([B.shape[0], 25], rm)

# Declare index vars
i, j, k, l = get_index_vars(4)

# Define the MTTKRP computation
A[i, j] = B[i, k, l] * D[l, j] * C[k, j]

# Perform the MTTKRP computation and write the result to file
pt.write("A.tns", A)

When you run the above Python program, TACO will generate code under the hood that efficiently performs the computation in one shot. This lets TACO avoid materializing the intermediate Khatri-Rao product, thus reducing the amount of memory accesses and speeding up the computation.