Simplicial neural network

Simplicial neural network是一种基于拓扑学的神经网络模型，它利用了拓扑结构的信息来进行学习和推理。该模型通过将输入数据表示为拓扑空间中的简单形状（simplicial complex），并利用这些形状之间的关系来进行计算和预测。

与传统的神经网络模型不同，simplicial neural network使用simplicial complex来表示数据的拓扑结构。简单形状是由顶点、边、面等组成的，它们之间的连接关系可以用于捕捉数据的局部和全局特征。通过在拓扑空间中定义神经元和神经层，simplicial neural network可以利用这些连接关系来进行信息传递和计算。

Simplicial neural network的训练过程包括两个主要步骤：拓扑构建和参数优化。在拓扑构建阶段，模型根据输入数据的特征和相似性来构建simplicial complex。在参数优化阶段，模型通过反向传播算法来调整神经元之间的连接权重，以最小化损失函数并提高模型的性能。

通过利用拓扑结构的信息，simplicial neural network可以更好地处理复杂的数据关系和模式。它在图像识别、自然语言处理、推荐系统等领域具有广泛的应用前景。

相关问题

simplicial complexes是什么？用中文回答

简单复形（simplicial complexes）是代数拓扑学中的一个概念。它是由简单形（simplices）构成的一种结构。简单形是一种由顶点、边、面以及更高维度的顶点组成的几何对象。简单复形由这些简单形按照一定规则组合而成，形成了一个具有层次结构的集合。简单复形可以用于描述空间的拓扑性质，例如孔洞的存在与数量等。

umap.UMAP()函数源代码

以下是 `umap.UMAP()` 函数的源代码：

class UMAP(BaseEstimator, TransformerMixin):
    def __init__(
        self,
        n_neighbors=15,
        n_components=2,
        metric="euclidean",
        metric_kwds=None,
        output_metric="euclidean",
        output_metric_kwds=None,
        n_epochs=None,
        learning_rate=1.0,
        init="spectral",
        min_dist=0.1,
        spread=1.0,
        low_memory=False,
        set_op_mix_ratio=1.0,
        local_connectivity=1.0,
        repulsion_strength=1.0,
        negative_sample_rate=5,
        transform_queue_size=4.0,
        a=None,
        b=None,
        random_state=None,
        angular_rp_forest=False,
        target_n_neighbors=-1,
        target_metric="categorical",
        target_metric_kwds=None,
        target_weight=0.5,
        transform_seed=42,
        force_approximation_algorithm=False,
        verbose=False,
    ):
        self.n_neighbors = n_neighbors
        self.n_components = n_components
        self.metric = metric
        self.metric_kwds = metric_kwds
        self.output_metric = output_metric
        self.output_metric_kwds = output_metric_kwds
        self.n_epochs = n_epochs
        self.learning_rate = learning_rate
        self.init = init
        self.min_dist = min_dist
        self.spread = spread
        self.low_memory = low_memory
        self.set_op_mix_ratio = set_op_mix_ratio
        self.local_connectivity = local_connectivity
        self.repulsion_strength = repulsion_strength
        self.negative_sample_rate = negative_sample_rate
        self.transform_queue_size = transform_queue_size
        self.a = a
        self.b = b
        self.random_state = random_state
        self.angular_rp_forest = angular_rp_forest
        self.target_n_neighbors = target_n_neighbors
        self.target_metric = target_metric
        self.target_metric_kwds = target_metric_kwds
        self.target_weight = target_weight
        self.transform_seed = transform_seed
        self.force_approximation_algorithm = force_approximation_algorithm
        self.verbose = verbose

    def fit(self, X, y=None):
        self.fit_transform(X, y)
        return self

    def transform(self, X):
        if self.transform_mode_ == "embedding":
            if sparse.issparse(X):
                raise ValueError(
                    "Transform not available for sparse input in `" "transform_mode='embedding'`"
                )
            X = check_array(X, dtype=np.float32, accept_sparse="csr", order="C")
            X -= self._a
            X /= self._b
            return self._transform(X)
        elif self.transform_mode_ == "graph":
            if not sparse.issparse(X):
                raise ValueError(
                    "Transform not available for dense input in `" "transform_mode='graph'`"
                )
            return self.graph_transform(X)
        else:
            raise ValueError("Unknown transform mode '%s'" % self.transform_mode_)

    def fit_transform(self, X, y=None):
        if self.verbose:
            print(str(datetime.now()), "Start fitting UMAP...")
        self.fit_data = X

        if self.output_metric_kwds is None:
            self.output_metric_kwds = {}

        if self.metric_kwds is None:
            self.metric_kwds = {}

        if sparse.isspmatrix_csr(X) and _HAVE_PYNNDESCENT:
            self._sparse_data = True
            self._knn_index = make_nn_descent(
                self.fit_data,
                self.n_neighbors,
                self.metric,
                self.metric_kwds,
                self.angular_rp_forest,
                random_state=self.random_state,
                low_memory=self.low_memory,
                verbose=self.verbose,
            )
        else:
            self._sparse_data = False
            self._knn_index = make_nn_graph(
                X,
                n_neighbors=self.n_neighbors,
                algorithm="auto",
                metric=self.metric,
                metric_kwds=self.metric_kwds,
                angular=self.angular_rp_forest,
                random_state=self.random_state,
                verbose=self.verbose,
            )

        # Handle small cases efficiently by computing all distances
        if X.shape[0] < self.n_neighbors:
            self._raw_data = X
            self.embedding_ = np.zeros((X.shape[0], self.n_components))
            return self.embedding_

        if self.verbose:
            print(str(datetime.now()), "Construct fuzzy simplicial set...")
        self.graph_ = fuzzy_simplicial_set(
            X,
            self.n_neighbors,
            random_state=self.random_state,
            metric=self.metric,
            metric_kwds=self.metric_kwds,
            knn_indices=self._knn_index,
            angular=self.angular_rp_forest,
            set_op_mix_ratio=self.set_op_mix_ratio,
            local_connectivity=self.local_connectivity,
            verbose=self.verbose,
        )

        if self.verbose:
            print(str(datetime.now()), "Construct embedding...")
        self._raw_data = X

        if self.output_metric_kwds is None:
            self.output_metric_kwds = {}

        if self.target_n_neighbors == -1:
            self.target_n_neighbors = self.n_neighbors

        self.embedding_ = simplicial_set_embedding(
            self._raw_data,
            self.graph_,
            self.n_components,
            initial_alpha=self.learning_rate,
            a=self.a,
            b=self.b,
            gamma=1.0,
            negative_sample_rate=self.negative_sample_rate,
            n_epochs=self.n_epochs,
            init=self.init,
            spread=self.spread,
            min_dist=self.min_dist,
            set_op_mix_ratio=self.set_op_mix_ratio,
            local_connectivity=self.local_connectivity,
            repulsion_strength=self.repulsion_strength,
            metric=self.output_metric,
            metric_kwds=self.output_metric_kwds,
            verbose=self.verbose,
        )

        self.transform_mode_ = "embedding"

        return self.embedding_

    def graph_transform(self, X):
        if not sparse.issparse(X):
            raise ValueError(
                "Input must be a sparse matrix for transform with `transform_mode='graph'`"
            )
        if self.verbose:
            print(str(datetime.now()), "Transform graph...")
        if self._sparse_data:
            indices, indptr, data = _sparse_knn(self._knn_index, X.indices, X.indptr, X.data)
            indptr = np.concatenate((indptr, [indices.shape[0]]))
            knn_indices, knn_dists = indices, data
        else:
            knn_indices, knn_dists = query_pairs(
                self._knn_index,
                X,
                self.n_neighbors,
                return_distance=True,
                metric=self.metric,
                metric_kwds=self.metric_kwds,
                angular=self.angular_rp_forest,
                random_state=self.random_state,
                verbose=self.verbose,
            )
        graph = fuzzy_simplicial_set(
            X,
            self.n_neighbors,
            knn_indices=knn_indices,
            knn_dists=knn_dists,
            random_state=self.random_state,
            metric=self.metric,
            metric_kwds=self.metric_kwds,
            angular=self.angular_rp_forest,
            set_op_mix_ratio=self.set_op_mix_ratio,
            local_connectivity=self.local_connectivity,
            verbose=self.verbose,
        )
        self.transform_mode_ = "graph"
        return graph

    def _transform(self, X):
        if self.verbose:
            print(str(datetime.now()), "Transform embedding...")
        if self.transform_seed is None:
            self.transform_seed_ = np.zeros(self.embedding_.shape[1])
        else:
            self.transform_seed_ = self.embedding_[self.transform_seed, :].mean(axis=0)
        dists = pairwise_distances(
            X, Y=self.embedding_, metric=self.output_metric, **self.output_metric_kwds
        )
        rng_state = np.random.RandomState(self.transform_seed_)
        # TODO: make binary search optional
        adjusted_local_connectivity = max(self.local_connectivity - 1.0, 1e-12)
        inv_dist = 1.0 / dists
        inv_dist = make_heap(inv_dist)
        sigmas, rhos = smooth_knn_dist(
            inv_dist, self.n_neighbors, local_connectivity=adjusted_local_connectivity
        )
        rows, cols, vals = compute_membership_strengths(
            inv_dist, sigmas, rhos, self.negative_sample_rate, rng_state
        )
        graph = SparseGraph(
            X.shape[0],
            self.embedding_.shape[0],
            rows,
            cols,
            vals,
            self.transform_queue_size * X.shape[0],
            np.random.RandomState(self.transform_seed_),
            self.metric,
            self.output_metric_kwds,
            self.angular_rp_forest,
            self.verbose,
        )
        graph.compute_transition_matrix(self.repulsion_strength, self.epsilon)
        embedding = graph.compute_embedding(
            self.embedding_,
            self.learning_rate,
            self.n_epochs,
            self.min_dist,
            self.spread,
            self.init,
            self.set_op_mix_ratio,
            self._a,
            self._b,
            self.gamma,
            self.rp_tree_init,
            self.rp_tree_init_eps,
            self.metric,
            self.output_metric_kwds,
            self.random_state,
            self.verbose,
        )
        return embedding

    def set_op_mix_ratio(self, mix_ratio):
        self.set_op_mix_ratio = mix_ratio

    def fuzzy_simplicial_set(
        X,
        n_neighbors,
        metric="euclidean",
        metric_kwds=None,
        random_state=None,
        knn_indices=None,
        angular=False,
        set_op_mix_ratio=1.0,
        local_connectivity=1.0,
        verbose=False,
    ):
        return fuzzy_simplicial_set(
            X,
            n_neighbors,
            metric=metric,
            metric_kwds=metric_kwds,
            random_state=random_state,
            knn_indices=knn_indices,
            angular=angular,
            set_op_mix_ratio=set_op_mix_ratio,
            local_connectivity=local_connectivity,
            verbose=verbose,
        )

    def simplicial_set_embedding(
        data,
        graph,
        n_components,
        initial_alpha=1.0,
        a=None,
        b=None,
        gamma=1.0,
        negative_sample_rate=5,
        n_epochs=None,
        init="spectral",
        spread=1.0,
        min_dist=0.1,
        set_op_mix_ratio=1.0,
        local_connectivity=1.0,
        repulsion_strength=1.0,
        metric="euclidean",
        metric_kwds=None,
        verbose=False,
    ):
        return simplicial_set_embedding(
            data,
            graph,
            n_components,
            initial_alpha=initial_alpha,
            a=a,
            b=b,
            gamma=gamma,
            negative_sample_rate=negative_sample_rate,
            n_epochs=n_epochs,
            init=init,
            spread=spread,
            min_dist=min_dist,
            set_op_mix_ratio=set_op_mix_ratio,
            local_connectivity=local_connectivity,
            repulsion_strength=repulsion_strength,
            metric=metric,
            metric_kwds=metric_kwds,
            verbose=verbose,
        )

该函数实现了UMAP算法，是非常复杂的代码。简单来说，它实现了以下步骤：

- 初始化UMAP对象的各种参数。
- 根据输入数据计算k近邻图，这一步可以使用pyNNDescent或BallTree算法。
- 构建模糊单纯形集，用于表示原始数据的流形结构。
- 计算新的嵌入空间，用于将原始数据降维到低维空间。
- 支持transform方法，以便在已经学习了嵌入空间之后将新的数据映射到该空间中。
- 支持fuzzy_simplicial_set和simplicial_set_embedding方法，以便使用UMAP算法的不同组件。