Add trace_dot in linalg (#4176)

* Implemented trace_dot in linalg
* Use trace_dot in LMNN

src/shogun/mathematics/linalg/LinalgNamespace.h

@@ -1690,6 +1690,19 @@ namespace shogun
 			return infer_backend(A)->transpose_matrix(A);
 		}
 
+		/**
+		 * Method that computes the trace of \f$AB\f$ as \f$sum(A.*B')$\f
+		 *
+		 * @param A The matrix A
+		 * @param B The matrix B
+		 * @return The trace of the product of A and B
+		 */
+		template <typename T>
+		T trace_dot(const SGMatrix<T>& A, const SGMatrix<T>& B)
+		{
+			return sum(element_prod(A, B, false, true));
+		}
+
 		/**
 		 * Solve the linear equations \f$Lx=b\f$,
 		 * where \f$L\f$ is a triangular matrix.

src/shogun/metric/LMNN.cpp

@@ -104,13 +104,10 @@ void CLMNN::train(SGMatrix<float64_t> init_transform)
 
 		// Compute the objective, trace of Mahalanobis distance matrix (L squared) times the gradient
 		// plus the number of current impostors to account for the margin
-		// trace of the product of two matrices computed fast using
-		// trace(A*B)=sum(A.*B')
 		SG_DEBUG("Computing objective.\n")
 		obj[iter] = m_regularization * cur_impostors.size();
-		obj[iter] += linalg::trace(
-		    linalg::element_prod(
-		        linalg::matrix_prod(L, L, true, false), gradient));
+		obj[iter] +=
+		    linalg::trace_dot(linalg::matrix_prod(L, L, true, false), gradient);
 
 		// Correct step size
 		CLMNNImpl::correct_stepsize(stepsize, obj, iter);

tests/unit/mathematics/linalg/operations/Eigen3_operations_unittest.cc

@@ -1734,6 +1734,23 @@ TEST(LinalgBackendEigen, SGMatrix_trace)
 	EXPECT_NEAR(trace(A), tr, 1e-15);
 }
 
+TEST(LinalgBackendEigen, SGMatrix_trace_dot)
+{
+	const index_t m = 2;
+	float64_t data_A[] = {0.68764958, 0.11456779, 0.75164207, 0.50436194};
+	float64_t data_B[] = {0.30786772, 0.25503552, 0.34367041, 0.66491478};
+
+	SGMatrix<float64_t> A(data_A, m, m, false);
+	SGMatrix<float64_t> B(data_B, m, m, false);
+
+	auto C = matrix_prod(A, B);
+	auto tr = 0.0;
+	for (auto i : range(m))
+		tr += C(i, i);
+
+	EXPECT_NEAR(tr, trace_dot(A, B), 1e-15);
+}
+
 TEST(LinalgBackendEigen, SGMatrix_transpose_matrix)
 {
 	const index_t m = 5, n = 3;