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Embedded Laplace: compute only the entries of A that compute_s2() reads #3472

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73 changes: 62 additions & 11 deletions stan/math/mix/functor/laplace_marginal_density.hpp
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Original file line number Diff line number Diff line change
Expand Up @@ -133,6 +133,35 @@ inline constexpr void copy_compute_s2(Output&& output, Input&& input) {
std::forward<Output>(output), std::forward<Input>(input));
}

/**
* Returns a `theta_size` x `theta_size` matrix that holds `entry(r, c)` in
* its diagonal blocks of size `hessian_block_size` and zero elsewhere.
* `laplace_likelihood::compute_s2()` reads only these entries of its matrix
* argument, so they are computed in O(n^2 b) instead of forming the full
* matrix with O(n^3) products.
* @tparam EntryFun Type of a functor `(Eigen::Index r, Eigen::Index c)`
* returning `double`
* @param theta_size Number of rows and columns
* @param hessian_block_size Size of the diagonal blocks
* @param entry Functor that returns the entry at row `r` and column `c`
*/
template <typename EntryFun>
inline Eigen::MatrixXd block_diagonal_entries(
const Eigen::Index theta_size, const Eigen::Index hessian_block_size,
EntryFun&& entry) {
Eigen::MatrixXd A = Eigen::MatrixXd::Zero(theta_size, theta_size);
const Eigen::Index n_blocks = theta_size / hessian_block_size;
for (Eigen::Index j = 0; j < n_blocks; ++j) {
const Eigen::Index start = j * hessian_block_size;
for (Eigen::Index c = start; c < start + hessian_block_size; ++c) {
for (Eigen::Index r = start; r < start + hessian_block_size; ++r) {
A(r, c) = entry(r, c);
}
}
}
return A;
}

} // namespace internal

/**
Expand Down Expand Up @@ -218,18 +247,23 @@ inline auto laplace_marginal_density(LLFun&& ll_fun, LLTupleArgs&& ll_args,
= md_est.L.template triangularView<Eigen::Lower>().solve(
md_est.W_r * covariance_val);
if constexpr (ll_args_contain_var) {
arena_t<Eigen::MatrixXd> A = covariance_val - C.transpose() * C;
// only the diagonal blocks of A = K - C' C, see compute_s2()
arena_t<Eigen::MatrixXd> A = internal::block_diagonal_entries(
covariance_val.rows(), options.hessian_block_size,
[&](Eigen::Index r, Eigen::Index c) {
return covariance_val(r, c) - C.col(r).dot(C.col(c));
});
auto s2_tmp = laplace_likelihood::compute_s2(
ll_fun, md_est.theta, A, options.hessian_block_size, ll_args_copy,
msgs);
s2.deep_copy(s2_tmp);
internal::copy_compute_s2<ZeroOut>(partial_parm, ll_args_filter);
} else {
s2.deep_copy(
(0.5
* (covariance_val.diagonal() - (C.transpose() * C).diagonal())
.cwiseProduct(laplace_likelihood::third_diff(
ll_fun, md_est.theta, ll_args_vals, msgs))));
s2.deep_copy((0.5
* (covariance_val.diagonal()
- C.colwise().squaredNorm().transpose())
.cwiseProduct(laplace_likelihood::third_diff(
ll_fun, md_est.theta, ll_args_vals, msgs))));
}

} else {
Expand All @@ -239,7 +273,12 @@ inline auto laplace_marginal_density(LLFun&& ll_fun, LLTupleArgs&& ll_args,
arena_t<Eigen::MatrixXd> C
= md_est.L.template triangularView<Eigen::Lower>().solve(
md_est.W_r * covariance_val);
arena_t<Eigen::MatrixXd> A = covariance_val - C.transpose() * C;
// only the diagonal blocks of A = K - C' C, see compute_s2()
arena_t<Eigen::MatrixXd> A = internal::block_diagonal_entries(
covariance_val.rows(), options.hessian_block_size,
[&](Eigen::Index r, Eigen::Index c) {
return covariance_val(r, c) - C.col(r).dot(C.col(c));
});
auto s2_tmp = laplace_likelihood::compute_s2(ll_fun, md_est.theta, A,
options.hessian_block_size,
ll_args_copy, msgs);
Expand All @@ -258,8 +297,14 @@ inline auto laplace_marginal_density(LLFun&& ll_fun, LLTupleArgs&& ll_args,
arena_t<Eigen::MatrixXd> C
= md_est.L.template triangularView<Eigen::Lower>().solve(
md_est.K_root.transpose());
// only the diagonal blocks of C' C, see compute_s2()
auto s2_tmp = laplace_likelihood::compute_s2(
ll_fun, md_est.theta, (C.transpose() * C).eval(),
ll_fun, md_est.theta,
internal::block_diagonal_entries(md_est.theta.size(),
options.hessian_block_size,
[&](Eigen::Index r, Eigen::Index c) {
return C.col(r).dot(C.col(c));
}),
options.hessian_block_size, ll_args_copy, msgs);
s2.deep_copy(s2_tmp);
internal::copy_compute_s2<ZeroOut>(partial_parm, ll_args_filter);
Expand All @@ -269,8 +314,14 @@ inline auto laplace_marginal_density(LLFun&& ll_fun, LLTupleArgs&& ll_args,
= Eigen::MatrixXd::Identity(md_est.W_r.rows(), md_est.W_r.cols())
- LU_solve_covariance * md_est.W_r;
R = md_est.W_r * I_minus_BinvKW; // == W - W B^{-1} K W
arena_t<Eigen::MatrixXd> A
= covariance_val - covariance_val * md_est.W_r * LU_solve_covariance;
// only the diagonal blocks of A = K - K W B^-1 K, see compute_s2()
const Eigen::MatrixXd KW = covariance_val * md_est.W_r;
arena_t<Eigen::MatrixXd> A = internal::block_diagonal_entries(
covariance_val.rows(), options.hessian_block_size,
[&](Eigen::Index r, Eigen::Index c) {
return covariance_val(r, c)
- KW.row(r).dot(LU_solve_covariance.col(c));
});
auto s2_tmp = laplace_likelihood::compute_s2(ll_fun, md_est.theta, A,
options.hessian_block_size,
ll_args_copy, msgs);
Expand All @@ -296,7 +347,7 @@ inline auto laplace_marginal_density(LLFun&& ll_fun, LLTupleArgs&& ll_args,
internal::collect_adjoints<ZeroOut>(partial_parm, ll_args_filter);
arena_t<Eigen::VectorXd> v;
if (md_est.solver_used == 1 || md_est.solver_used == 2) {
v = covariance_val * s2 - covariance_val * R * covariance_val * s2;
v = covariance_val * s2 - covariance_val * (R * (covariance_val * s2));
} else {
v = LU_solve_covariance * s2;
}
Expand Down
72 changes: 72 additions & 0 deletions test/unit/math/laplace/laplace_block_diagonal_entries_test.cpp
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Original file line number Diff line number Diff line change
@@ -0,0 +1,72 @@
#include <stan/math/mix.hpp>
#include <gtest/gtest.h>
#include <ostream>
#include <tuple>
#include <vector>

namespace {

// symmetric test matrix without zero entries
Eigen::MatrixXd dense_test_matrix(int n) {
Eigen::MatrixXd m(n, n);
for (int c = 0; c < n; ++c) {
for (int r = 0; r < n; ++r) {
m(r, c) = 1.0 / (1.0 + r + c) + (r == c ? 2.0 : 0.0);
}
}
return m;
}

} // namespace

TEST(laplace_block_diagonal_entries, keeps_only_the_diagonal_blocks) {
const int n = 6;
const Eigen::MatrixXd full = dense_test_matrix(n);
for (int block_size : {1, 2, 3}) {
const Eigen::MatrixXd blocks = stan::math::internal::block_diagonal_entries(
n, block_size,
[&](Eigen::Index r, Eigen::Index c) { return full(r, c); });
ASSERT_EQ(n, blocks.rows());
ASSERT_EQ(n, blocks.cols());
for (int c = 0; c < n; ++c) {
for (int r = 0; r < n; ++r) {
if (r / block_size == c / block_size) {
EXPECT_EQ(full(r, c), blocks(r, c));
} else {
EXPECT_EQ(0.0, blocks(r, c));
}
}
}
}
}

// laplace_marginal_density() passes only the diagonal blocks of A to
// compute_s2(), because compute_s2() reads no other entry of A: the result
// is the same for the full matrix and for its diagonal blocks.
TEST(laplace_block_diagonal_entries,
compute_s2_reads_only_the_diagonal_blocks) {
const int n = 6;
const Eigen::MatrixXd full = dense_test_matrix(n);
Eigen::VectorXd theta(n);
theta << 0.3, -0.2, 0.5, 1.0, -0.7, 0.1;
const std::vector<int> y{1, 0, 2, 3, 0, 1, 2};
const std::vector<int> y_index{1, 2, 3, 4, 5, 6, 3};
const Eigen::VectorXd mean = Eigen::VectorXd::Constant(n, 0.4);
std::ostream* msgs = nullptr;
for (int block_size : {1, 2, 3}) {
const Eigen::MatrixXd blocks = stan::math::internal::block_diagonal_entries(
n, block_size,
[&](Eigen::Index r, Eigen::Index c) { return full(r, c); });
const Eigen::VectorXd s2_full = stan::math::laplace_likelihood::compute_s2(
stan::math::poisson_log_likelihood{}, theta, full, block_size,
std::forward_as_tuple(y, y_index, mean), msgs);
const Eigen::VectorXd s2_blocks
= stan::math::laplace_likelihood::compute_s2(
stan::math::poisson_log_likelihood{}, theta, blocks, block_size,
std::forward_as_tuple(y, y_index, mean), msgs);
ASSERT_EQ(n, s2_full.size());
for (int i = 0; i < n; ++i) {
EXPECT_EQ(s2_full(i), s2_blocks(i)) << "block size " << block_size;
}
}
}
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