Project: 049_ProtComposer_Compositional_Protein_Structure_Gener | Function: sample_blobs (iclr)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Statistical model for generating novel and diverse ellipsoid layouts using rejection sampling. Samples ellipsoid means from isotropic Gaussians, covariances from Wishart distributions, and uses Boltzmann factor to prevent overlaps, enabling diverse protein generation beyond training data.

**KEY TERMS:** {
  "nu": "degrees of freedom parameter ν",
  "psi": "scale parameter ψ",
  "sigma": "spread parameter σ",
  "score": "energy function U"
}

**CONTEXT CODE:** # SPDX-FileCopyrightText: Copyright (c) 2025 NVIDIA CORPORATION & AFFILIATES. All rights reserved.
# SPDX-License-Identifier: LicenseRef-NvidiaProprietary
#
# NVIDIA CORPORATION, its affiliates and licensors retain all intellectual
# property and proprietary rights in and to this material, related
# documentation and any modifications thereto. Any use, reproduction,
# disclosure or distribution of this material and related documentation
# without an express license agreement from NVIDIA CORPORATION or
# its affiliates is strictly prohibited.


import numpy as np
import networkx as nx
import numpy as np
from scipy.special import logsumexp, softmax
import torch
from torch.distributions.multivariate_normal import MultivariateNormal
from scipy.stats import wishart

alpha_slope = 0.22213567936924072
alpha_intercept = 7.512307373103329

beta_slope = 0.2727479880396384
beta_intercept = 8.28561006547174


def blobify(pos, dssp, radius_thresh=5, size_thresh=5):
    raise NotImplementedError("Function implementation has been masked for testing")

def shannon_complexity(pos, dssp, thresh=5):
    blobs = blobify(pos, dssp, radius_thresh=thresh)
    counts = np.array([b["count"] for b in blobs])
    probs = counts / counts.sum()
    return -(probs * np.log(probs)).sum()


def score_blobs(centers, covars):
    energy = 0
    for pos, cov in zip(centers, covars):
        relpos = centers - pos
        d = np.einsum("li,ij,lj->l", relpos, np.linalg.inv(cov), relpos) ** 0.5
        d = d[d != 0]
        energy += (1 / d**2).sum()
    return energy


def sample_blobs(k, nu, psi, sigma):
    raise NotImplementedError("Function implementation has been masked for testing")

def gmm_ll(pos, centers, covars, probs):
    probs = probs / probs.sum()
    relpos = pos[:, None] - centers[None]
    ll = (
        -0.5 * np.einsum("lni,nij,lnj->ln", relpos, np.linalg.inv(covars), relpos)
        - 0.5 * np.linalg.slogdet(covars)[1][None]
        + np.log(probs)[None]
    )
    return logsumexp(ll, -1)


def dssp_gmm_ll(pos, centers, covars, probs, pos_dssp, centers_dssp):
    if np.any(centers_dssp == 1):
        gmm_ll_1 = gmm_ll(
            pos,
            centers[centers_dssp == 1],
            covars[centers_dssp == 1],
            probs[centers_dssp == 1],
        )
        gmm_ll_1[pos_dssp != 1] = -np.inf
    else:
        gmm_ll_1 = -np.inf * np.ones(len(pos))

    if np.any(centers_dssp == 2):
        gmm_ll_2 = gmm_ll(
            pos,
            centers[centers_dssp == 2],
            covars[centers_dssp == 2],
            probs[centers_dssp == 2],
        )
        gmm_ll_2[pos_dssp != 2] = -np.inf
    else:
        gmm_ll_2 = -np.inf * np.ones(len(pos))

    dssp_1_count = probs[centers_dssp == 1].sum() / probs.sum()
    dssp_2_count = probs[centers_dssp == 2].sum() / probs.sum()

    return np.log(np.exp(gmm_ll_1) * dssp_1_count + np.exp(gmm_ll_2) * dssp_2_count)


"""
Treats the blobs as a GMM and evaluates the average log-DENSITY at each pos.
Note that DENSITY is N * PDF where N is the total number of residues in the blobs.
Not suitable for partial blobs.
"""


def blob_likelihood(pos, dssp, blobs, structured_only=False, mean=True):
    if structured_only:
        pos = pos[dssp != 0]
        dssp = dssp[dssp != 0]
        if len(dssp) == 0:
            return -10
    if len(blobs) == 0:
        ll = -np.inf * np.ones(pos.shape[0])
    else:
        covars = np.stack([blob["covar"] for blob in blobs])
        counts = np.array([blob["count"] for blob in blobs])
        centers = np.stack([blob["pos"] for blob in blobs])

        ll = gmm_ll(pos, centers, covars, counts)
        ll += np.log(counts.sum())

    if not mean:
        return ll
    return ll.mean()


"""
Reblobs the protein and evaluates the number of residues per blob
"""


def blobs_per_res(pos, dssp, thresh=5):
    blobs = blobify(pos, dssp, thresh)
    return len(blobs) / len(pos)


def res_per_blob(pos, dssp, thresh=5):
    blobs = blobify(pos, dssp, thresh)
    return len(pos) / len(blobs)


"""
Reblobs the protein and evaluates the JSD between the GMMs 
defined by the original and new blobs.
Not suitable for partial blobs.
If use_dssp=True, consider sample space R^3 x {1,2}
"""


def reblob_jsd(pos, dssp, blobs, num_samples=1000, use_dssp=False, thresh=5):
    p_blobs = blobs
    q_blobs = blobify(pos, dssp, thresh)

    p_covars = np.stack([blob["covar"] for blob in p_blobs])
    p_counts = np.array([blob["count"] for blob in p_blobs])


Complete the function implementation replacing "raise NotImplementedError".