def lambda_sum_largest(X, k):
"""Sum of the largest k eigenvalues.
"""
X = Expression.cast_to_const(X)
if X.size[0] != X.size[1]:
raise ValueError("First argument must be a square matrix.")
elif int(k) != k or k <= 0:
raise ValueError("Second argument must be a positive integer.")
"""
S_k(X) denotes lambda_sum_largest(X, k)
t >= k S_k(X - Z) + trace(Z), Z is PSD
implies
t >= ks + trace(Z)
Z is PSD
sI >= X - Z (PSD sense)
which implies
t >= ks + trace(Z) >= S_k(sI + Z) >= S_k(X)
We use the fact that
S_k(X) = sup_{sets of k orthonormal vectors u_i}\sum_{i}u_i^T X u_i
and if Z >= X in PSD sense then
\sum_{i}u_i^T Z u_i >= \sum_{i}u_i^T X u_i
We have equality when s = lambda_k and Z diagonal
with Z_{ii} = (lambda_i - lambda_k)_+
"""
Z = Semidef(X.size[0])
return k*lambda_max(X - Z) + trace(Z)
def lambda_sum_largest(X, k):
"""Sum of the largest k eigenvalues.
"""
X = Expression.cast_to_const(X)
if X.size[0] != X.size[1]:
raise ValueError("First argument must be a square matrix.")
elif int(k) != k or k <= 0:
raise ValueError("Second argument must be a positive integer.")
"""
S_k(X) denotes lambda_sum_largest(X, k)
t >= k S_k(X - Z) + trace(Z), Z is PSD
implies
t >= ks + trace(Z)
Z is PSD
sI >= X - Z (PSD sense)
which implies
t >= ks + trace(Z) >= S_k(sI + Z) >= S_k(X)
We use the fact that
S_k(X) = sup_{sets of k orthonormal vectors u_i}\sum_{i}u_i^T X u_i
and if Z >= X in PSD sense then
\sum_{i}u_i^T Z u_i >= \sum_{i}u_i^T X u_i
We have equality when s = lambda_k and Z diagonal
with Z_{ii} = (lambda_i - lambda_k)_+
"""
Z = Semidef(X.size[0])
return k*lambda_max(X - Z) + trace(Z)
def lambda_min(X):
""" Minimum eigenvalue; :math:`\\lambda_{\\min}(A)`.
"""
X = Expression.cast_to_const(X)
return -lambda_max(-X)
