eMaTe can run in both CPU and GPU and can estimate the spectral density and related trace functions, such as entropy and Estrada index, even in matrices (directed or undirected graphs) with million of nodes.

## Project description

eMaTe it is a python package which the main goal is to provide methods capable of estimating the spectral densities and trace functions of large sparse matrices. eMaTe can run in both CPU and GPU and can estimate the spectral density and related trace functions, such as entropy and Estrada index, even in directed or undirected networks with million of nodes.

## Install

```
pip install emate
```

If you a have a GPU you should also install cupy.

## Kernel Polynomial Method (KPM)

The Kernel Polynomial Method can estimate the spectral density of large sparse Hermitan matrices with a computational cost almost linear. This method combines three key ingredients: the Chebyshev expansion + the stochastic trace estimator + kernel smoothing.

### Example

import networkx as nx import numpy as np n = 3000 g = nx.erdos_renyi_graph(n , 3/n) W = nx.adjacency_matrix(g) vals = np.linalg.eigvals(W.todense()).real

from emate.hermitian import tfkpm num_moments = 40 num_vecs = 40 extra_points = 10 ek, rho = tfkpm(W, num_moments, num_vecs, extra_points)

import matplotlib.pyplot as plt plt.hist(vals, density=True, bins=100, alpha=.9, color="steelblue") plt.scatter(ek, rho, c="tomato", zorder=999, alpha=0.9, marker="d")

If the CUPY package it is available in your machine, you can also use the cupy implementation. When compared to tf-kpm, the Cupy-kpm is slower for median matrices (100k) and faster for larger matrices (> 10^6). The main reason it's because the tf-kpm was implemented in order to calc all te moments in a single step.

import matplotlib.pyplot as plt from emate.hermitian import cupykpm num_moments = 40 num_vecs = 40 extra_points = 10 ek, rho = cupykpm(W.tocsr(), num_moments, num_vecs, extra_points) plt.hist(vals, density=True, bins=100, alpha=.9, color="steelblue") plt.scatter(ek.get(), rho.get(), c="tomato", zorder=999, alpha=0.9, marker="d")

## Stochastic Lanczos Quadrature (SLQ)

The problem of estimating the trace of matrix functions appears in applications ranging from machine learning and scientific computing, to computational biology.[2]

### Example

#### Computing the Estrada index

from emate.symmetric.slq import pyslq import tensorflow as tf def trace_function(eig_vals): return tf.exp(eig_vals) num_vecs = 100 num_steps = 50 approximated_estrada_index, _ = pyslq(L_sparse, num_vecs, num_steps, trace_function) exact_estrada_index = np.sum(np.exp(vals_laplacian)) approximated_estrada_index, exact_estrada_index

The above code returns

```
(3058.012, 3063.16457163222)
```

#### Entropy

import scipy import scipy.sparse def entropy(eig_vals): s = 0. for val in eig_vals: if val > 0: s += -val*np.log(val) return s L = np.array(G.laplacian(normalized=True), dtype=np.float64) vals_laplacian = np.linalg.eigvalsh(L).real exact_entropy = entropy(vals_laplacian) def trace_function(eig_vals): def entropy(val): return tf.cond(val>0, lambda:-val*tf.log(val), lambda: 0.) return tf.map_fn(entropy, eig_vals) L_sparse = scipy.sparse.coo_matrix(L) num_vecs = 100 num_steps = 50 approximated_entropy, _ = pyslq(L_sparse, num_vecs, num_steps, trace_function) approximated_entropy, exact_entropy

```
(-509.46283, -512.5283224633046)
```

[3] The Kernel Polynomial Method applied to tight binding systems with time-dependence

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