拓扑机器学习的神圣三件套：Gudhi，Scikit-Learn和Tensorflow（附链接&代码）

TDA：

https://en.wikipedia.org/wiki/Topological_data_analysis

IBM：

https://researcher.watson.ibm.com/researcher/view_group.php?id=6585

Fujitsu：

https://www.fujitsu.com/global/about/resources/news/press-releases/2016/0216-01.html

Ayasdi：

https://www.ayasdi.com/platform/technology/

生物学：

https://www.ncbi.nlm.nih.gov/pubmed/28459448

时间序列：

https://www.ams.org/journals/notices/201905/rnoti-p686.pdf

金融：

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2931836

科学可视化：

https://topology-tool-kit.github.io/

计算机图形学：

http://www.lix.polytechnique.fr/~maks/papers/top_opt_SGP18.pdf

TDA的参考示例：点云分类

这个数据集在一篇开创性的TDA文章上介绍过。它由通过下述动力系统生成的轨迹来得到的点云集组成：

球并集的例子。对于中间图的并集，它清晰的组成了一个洞。整张图片被我“不要脸”地借用自我之前的一个帖子

import gudhirips = gudhi.RipsComplex(points=X).create_simplex_tree()


    
dgm = rips.persistence()

这个漂亮的持续图由r=4.1对应的点云计算出。红色的点代表相连的成分，蓝色的点代表洞

接下来我们将解决的任务则是给定点云预测r的值。

表达

https://gudhi.inria.fr/python/latest/representations.html

import gudhi.representations as tdafrom sklearn.pipeline import Pipelinefrom sklearn.svm import SVCfrom sklearn.ensemble import RandomForestClassifier as RFfrom sklearn.neighbors import KNeighborsClassifier as kNNfrom sklearn.model_selection import GridSearchCV

pipe = Pipeline([("TDA",       tda.PersistenceImage()),                 ("Estimator", SVC())])param =    [{"TDA": [tda.SlicedWassersteinKernel()],              "TDA__bandwidth": [0.1, 1.0],             "TDA__num_directions": [20],             "Estimator": [SVC(kernel="precomputed")]},                        {"TDA": [tda.PersistenceWeightedGaussianKernel()],              "TDA__bandwidth": [0.1, 0.01],             "TDA__weight": [lambda x: np.arctan(x[1]-x[0])],              "Estimator": [SVC(kernel="precomputed")]},                        {"TDA": [tda.PersistenceImage()],              "TDA__resolution": [ [5,5], [6,6] ],             "TDA__bandwidth": [0.01, 0.1, 1.0, 10.0],             "Estimator": [SVC()]},                        {"TDA": [tda.Landscape()],              "TDA__resolution": [100],             "Estimator": [RF()]},                       {"TDA": [tda.BottleneckDistance()],              "TDA__epsilon": [0.1],              "Estimator: [kNN(metric="precomputed")]}           ]model = GridSearchCV(pipe, param, cv=3)model = model.fit(diagrams, labels)

带切片Wasserstein核：

http://proceedings.mlr.press/v70/carriere17a/carriere17a.pdf

持续权重Gaussian核：

http://proceedings.mlr.press/v48/kusano16.html

Persistence Images：

http://jmlr.org/papers/v18/16-337.html

Persistence Landscapes：

http://www.jmlr.org/papers/volume16/bubenik15a/bubenik15a.pdf

Gudhi的Tutorial：

https://github.com/GUDHI/TDA-tutorial/blob/master/Tuto-GUDHI-representations.ipynb

Gudhi（1）：

http://gudhi.gforge.inria.fr/python/latest/

Gudhi（2）：

https://gudhi.inria.fr/python/latest/

这篇论文：

https://sites.google.com/view/hiraoka-lab-en/research/mathematical-research/continuation-of-point-cloud-data-via-persistence-diagram

https://github.com/GUDHI/TDA-tutorial/blob/master/Tuto-GUDHI-optimization.ipynb

def compute_rips(x):    rc = gd.RipsComplex(points=x)    st = rc.create_simplex_tree()    dgm = st.persistence()    pairs = st.persistence_pairs()    return [dgm, pairs]

def compute_rips_grad(grad_dgm, pairs, x):    grad_x = np.zeros(x.shape, dtype=np.float32)    for i in range(len(dgm)):            [v0a, v0b] = pairs[i][0]            [v1a, v1b] = pairs[i][1]            grad_x[v0a,:]+=grad_dgm[i,0]*(x[v0a,:]-x[v0b,:])/val0            grad_x[v0b,:]+=grad_dgm[i,0]*(x[v0b,:]-x[v0a,:])/val0            grad_x[v1a,:]+=grad_dgm[i,1]*(x[v1a,:]-x[v1b,:])/val1            grad_x[v1b,:]+=grad_dgm[i,1]*(x[v1b,:]-x[v1a,:])/val1    return grad_x

import tensorflow as tffrom tensorflow.python.framework import opsdef py_func(func, inp, Tout, stateful=True, name=None, grad=None):    rnd_name = "PyFuncGrad" + str(np.random.randint(0, 1e+8))    tf.RegisterGradient(rnd_name)(grad)    g = tf.get_default_graph()    with g.gradient_override_map({"PyFunc": rnd_name}):        return tf.py_func(func, inp, Tout, stateful=stateful, name=name)def Rips(card, hom_dim, x, Dx, max_length, name=None ):    with ops.op_scope([x], name, "Rips") as name:        return py_func(compute_rips, [x], [tf.float32], name=name, grad=_RipsGrad)def _RipsGrad(op, grad_dgm):    pairs = op.outputs[1]    x = op.inputs[0]    grad_x = tf.py_func(compute_rips_grad, [grad_dgm,pairs,x], [tf.float32])[0]    return [None, None, grad_x, None, None]

tf.reset_default_graph()x = tf.get_variable("X", shape=[n_pts,2], initializer=tf.random_uniform_initializer(0.,1.), trainable=True)dgm, pairs = Rips(x)loss = -tf.reduce_sum(tf.square(dgm[:,1]-dgm[:,0]))opt = tf.train.GradientDescentOptimizer(learning_rate=0.1)train = opt.minimize(loss)

https://github.com/GUDHI/TDA-tutorial/blob/master/Tuto-GUDHI-optimization.ipynb

译者简介：孙韬淳，首都师范大学大四在读，主修遥感科学与技术。目前专注于基本知识的掌握和提升，期望在未来有机会探索数据科学在地学应用的众多可能性。爱好之一为翻译创作，在业余时间加入到THU数据派平台的翻译志愿者小组，希望能和大家一起交流分享，共同进步。

「完」

转自：数据派THU ；

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