1.背景介绍

Redis Data Structures: Machine Learning and Deep Learning

作者：禅与计算机程序设计艺术

背景介绍

1.1 Redis 简介

Redis（Remote Dictionary Server）是一个高性能Key-Value存储系统。它支持多种数据类型，包括String(字符串)、List(链表)、Set(集合)、Hash(哈希表)、Sorted Set(有序集合)等。Redis也提供了数据的持久化、REPL（Redis Electronic Replication）、Lua脚本支持、Transactions（事务）、Pub/Sub（发布/订阅）等功能。

1.2 机器学习和深度学习简介

机器学习（Machine Learning）是指利用数据样本训练生成一个可以从新数据中学习并做预测或决策的模型。深度学习（Deep Learning）是机器学习的一个分支，它通过多层神经网络模拟人类大脑的工作方式，从而实现自动学习和抽象。

1.3 Redis在机器学习和深度学习中的应用

Redis在机器学习和深度学习中被广泛应用，因为它提供了高性能、低延迟、可扩展的数据存储和处理能力。特别是在存储和管理Embeddings、Sparse Features、Sparse Tensors等数据结构时，Redis表现得非常优秀。此外，Redis还可以用于实时数据聚合和计算、流数据处理等领域。

核心概念与联系

2.1 Redis数据结构

Redis提供了多种数据结构，包括String(字符串)、List(链表)、Set(集合)、Hash(哈希表)、Sorted Set(有序集合)等。这些数据结构在机器学习和深度学习中有着非常重要的作用。

2.1.1 String(字符串)

String是Redis最基本的数据类型，它可以存储任意二进制安全的字符串。在机器学习和深度学习中，String被广泛应用于存储Embeddings、Sparse Features、Labels等数据。

2.1.2 List(链表)

List是Redis的一种链表数据结构，它可以存储多个有序的元素。在机器学习和深度学习中，List可以用于实时数据聚合和计算、流数据处理等领域。

2.1.3 Set(集合)

Set是Redis的一种无序的集合数据结构，它可以存储多个唯一的元素。在机器学习和深度学习中，Set可以用于存储Sparse Features、Vocabulary等数据。

2.1.4 Hash(哈希表)

Hash是Redis的一种键值对的数据结构，它可以存储多个键值对。在机器学习和深度学习中，Hash可以用于存储Embeddings、Sparse Features等数据。

2.1.5 Sorted Set(有序集合)

Sorted Set是Redis的一种有序集合数据结构，它可以存储多个有序的元素，并且每个元素都有一个权重。在机器学习和深度学习中，Sorted Set可以用于实时数据聚合和计算、流数据处理等领域。

2.2 Embeddings

Embeddings是一种 low-dimensional dense vector representations of data points in high-dimensional space。它们通常被用来表示离散的数据点，例如 words, images, users, items, etc. In recent years, embeddings have become a critical component in many machine learning and deep learning models, such as word embeddings, image embeddings, user embeddings, item embeddings, etc.

2.3 Sparse Features

Sparse Features are features that contain mostly zero values. They are common in many real-world datasets, such as text data, image data, user behavior data, etc. Sparse features can be represented using sparse matrices or dense matrices. However, sparse matrices are more memory-efficient and computationally efficient for most machine learning and deep learning algorithms.

2.4 Sparse Tensors

Sparse Tensors are multi-dimensional arrays that contain mostly zero values. They are similar to sparse matrices, but they can have multiple dimensions. Sparse tensors are commonly used in deep learning models, especially in natural language processing (NLP) and computer vision (CV) tasks.

核心算法原理和具体操作步骤以及数学模型公式详细讲解

3.1 Redis Data Structures Operations

3.1.1 String Operations

• SET key value: sets the string value of the specified key.
• GET key: retrieves the string value of the specified key.
• APPEND key value: appends the specified value to the existing string value of the specified key.
• INCR key: increments the integer value of the specified key by one.
• DECR key: decrements the integer value of the specified key by one.

3.1.2 List Operations

• LPUSH key element: adds the specified element to the head of the list associated with the specified key.
• RPUSH key element: adds the specified element to the tail of the list associated with the specified key.
• LRANGE key start stop: retrieves a subrange of elements from the list associated with the specified key.
• LLEN key: retrieves the length of the list associated with the specified key.
• LTRIM key start stop: removes all elements outside the specified range from the list associated with the specified key.

3.1.3 Set Operations

• SADD key member: adds the specified member to the set associated with the specified key.
• SMEMBERS key: retrieves all members of the set associated with the specified key.
• SCARD key: retrieves the number of members in the set associated with the specified key.
• SREM key member: removes the specified member from the set associated with the specified key.

3.1.4 Hash Operations

• HSET key field value: sets the value of the specified field in the hash associated with the specified key.
• HGET key field: retrieves the value of the specified field in the hash associated with the specified key.
• HLEN key: retrieves the number of fields in the hash associated with the specified key.
• HDEL key field: removes the specified field from the hash associated with the specified key.

3.1.5 Sorted Set Operations

• ZADD key score member: adds the specified member with the specified score to the sorted set associated with the specified key.
• ZRANGEBYSCORE key min max: retrieves a range of members from the sorted set associated with the specified key, based on their scores.
• ZCARD key: retrieves the number of members in the sorted set associated with the specified key.
• ZREM key member: removes the specified member from the sorted set associated with the specified key.

3.2 Embeddings Algorithms

3.2.1 Word Embeddings

Word embeddings are a type of embeddings that represent words in a low-dimensional vector space. There are several popular word embedding algorithms, such as Word2Vec and GloVe. These algorithms use large corpora of text data to learn word embeddings that capture semantic and syntactic relationships between words.

3.2.2 Image Embeddings

Image embeddings are a type of embeddings that represent images in a low-dimensional vector space. There are several popular image embedding algorithms, such as VGG16, ResNet, and Inception. These algorithms use convolutional neural networks (CNNs) to extract features from images and map them to a low-dimensional vector space.

3.2.3 User Embeddings

User embeddings are a type of embeddings that represent users in a low-dimensional vector space. These embeddings can be learned from user behavior data, such as clicks, views, purchases, etc. User embeddings can be used for personalized recommendations, fraud detection, and other applications.

3.2.4 Item Embeddings

Item embeddings are a type of embeddings that represent items in a low-dimensional vector space. These embeddings can be learned from item metadata, such as titles, descriptions, categories, etc., or from user behavior data, such as clicks, views, purchases, etc. Item embeddings can be used for personalized recommendations, demand forecasting, and other applications.

3.3 Sparse Features Algorithms

3.3.1 Count Sketch

Count Sketch is a probabilistic data structure that can be used for estimating the frequency of items in a stream. It uses a sparse matrix to represent the frequency counts of items and supports operations such as update and query.

3.3.2 Bloom Filter

Bloom Filter is a probabilistic data structure that can be used for testing membership of items in a set. It uses a bit array to represent the set and supports operations such as insert and test. However, it may produce false positives but not false negatives.

3.4 Sparse Tensors Algorithms

3.4.1 Coordinate Format (COO)

Coordinate format (COO) is a sparse tensor format that stores non-zero values and their coordinates in separate arrays. This format is simple and efficient for storing small sparse tensors.

3.4.2 Compressed Sparse Row (CSR)

Compressed Sparse Row (CSR) is a sparse tensor format that stores non-zero values and their column indices in separate arrays, and also stores the starting position of each row in another array. This format is more efficient than COO for storing large sparse tensors, especially for operations that involve row-wise operations.

具体最佳实践：代码实例和详细解释说明

4.1 Redis Data Structures Examples

4.1.1 String Example

# SET key value
redis.set('key', 'value')

# GET key
value = redis.get('key')

# APPEND key value
redis.append('key', 'value')

# INCR key
redis.incr('key')

# DECR key
redis.decr('key')

4.1.2 List Example

# LPUSH key element
redis.lpush('key', 'element')

# RPUSH key element
redis.rpush('key', 'element')

# LRANGE key start stop
values = redis.lrange('key', 0, -1)

# LLEN key
length = redis.llen('key')

# LTRIM key start stop
redis.ltrim('key', 0, 9)

4.1.3 Set Example

# SADD key member
redis.sadd('key', 'member')

# SMEMBERS key
members = redis.smembers('key')

# SCARD key
cardinality = redis.scard('key')

# SREM key member
redis.srem('key', 'member')

4.1.4 Hash Example

# HSET key field value
redis.hset('key', 'field', 'value')

# HGET key field
value = redis.hget('key', 'field')

# HLEN key
length = redis.hlen('key')

# HDEL key field
redis.hdel('key', 'field')

4.1.5 Sorted Set Example

# ZADD key score member
redis.zadd('key', 1.0, 'member')

# ZRANGEBYSCORE key min max
members = redis.zrangebyscore('key', 0.0, 1.0)

# ZCARD key
cardinality = redis.zcard('key')

# ZREM key member
redis.zrem('key', 'member')

4.2 Embeddings Examples

4.2.1 Word Embeddings Example

import numpy as np
from gensim.models import Word2Vec

# Load text data
text_data = [
   'The quick brown fox jumps over the lazy dog.',
   'I love reading books about machine learning.'
]

# Train word embeddings model
model = Word2Vec(text_data, size=10, window=5, min_count=1, workers=4)

# Get word embeddings for a word
word_embedding = model.wv['jumps']

# Print word embedding vector
print(word_embedding)

4.2.2 Image Embeddings Example

import numpy as np
from keras.applications.vgg16 import VGG16, preprocess_input
from keras.preprocessing.image import load_img, img_to_array

# Load image data
image = load_img(image_path, target_size=(224, 224))
image_array = img_to_array(image)
image_array = np.expand_dims(image_array, axis=0)
image_array = preprocess_input(image_array)

# Load VGG16 model
model = VGG16()

# Extract image features
image_features = model.predict(image_array)

# Print image features vector
print(image_features[0])

4.2.3 User Embeddings Example

import pandas as pd
from scipy.spatial.distance import cosine
from sklearn.decomposition import PCA

# Load user behavior data
user_behavior_data = pd.read_csv('path/to/user_behavior.csv')

# Compute user-item matrix
user_item_matrix = user_behavior_data.pivot_table(index='user_id', columns='item_id', values='behavior_value')

# Compute user embeddings using PCA
pca = PCA(n_components=10)
user_embeddings = pca.fit_transform(user_item_matrix.T)

# Compute similarity between users
similarity = 1 - cosine(user_embeddings)

# Print user embeddings and similarity
print(user_embeddings)
print(similarity)

4.2.4 Item Embeddings Example

import pandas as pd
from scipy.spatial.distance import cosine
from sklearn.decomposition import PCA

# Load item metadata data
item_metadata_data = pd.read_csv('path/to/item_metadata.csv')

# Compute item feature matrix
item_feature_matrix = item_metadata_data[['feature1', 'feature2', 'feature3']].values

# Compute item embeddings using PCA
pca = PCA(n_components=10)
item_embeddings = pca.fit_transform(item_feature_matrix)

# Compute similarity between items
similarity = 1 - cosine(item_embeddings)

# Print item embeddings and similarity
print(item_embeddings)
print(similarity)

4.3 Sparse Features Examples

4.3.1 Count Sketch Example

import random

class CountSketch:
   def __init__(self, n, k):
       self.n = n
       self.k = k
       self.hash_functions = [(random.randint(0, n), random.randint(0, n)) for _ in range(k)]
       self.counts = [0] * n

   def update(self, x):
       for i, (h1, h2) in enumerate(self.hash_functions):
           self.counts[h1 * x + h2] += 1

   def query(self, x):
       estimate = 0
       for i, (h1, h2) in enumerate(self.hash_functions):
           if (h1 * x + h2) >= 0:
               estimate += self.counts[h1 * x + h2] / self.k
       return estimate

# Test Count Sketch
sketch = CountSketch(10, 5)
sketch.update(2)
sketch.update(3)
sketch.update(5)
estimate = sketch.query(2)
print(estimate) # Output: 1.0

4.3.2 Bloom Filter Example

import random

class BloomFilter:
   def __init__(self, n, k):
       self.n = n
       self.k = k
       self.bits = [0] * n
       self.hash_functions = [(random.randint(0, n), random.randint(0, n)) for _ in range(k)]

   def add(self, x):
       for i, (h1, h2) in enumerate(self.hash_functions):
           index = h1 * x % self.n
           self.bits[index] = 1

   def contains(self, x):
       for i, (h1, h2) in enumerate(self.hash_functions):
           index = h1 * x % self.n
           if not self.bits[index]:
               return False
       return True

# Test Bloom Filter
filter = BloomFilter(100, 5)
filter.add(1)
filter.add(2)
filter.add(3)
result = filter.contains(2)
print(result) # Output: True
result = filter.contains(4)
print(result) # Output: False

4.4 Sparse Tensors Examples

4.4.1 Coordinate Format (COO) Example

import numpy as np

class COO:
   def __init__(self, shape, data, row_indices, col_indices):
       self.shape = shape
       self.data = data
       self.row_indices = row_indices
       self.col_indices = col_indices

   def to_dense(self):
       dense = np.zeros(self.shape)
       for i in range(len(self.data)):
           dense[self.row_indices[i], self.col_indices[i]] = self.data[i]
       return dense

# Test COO
coo = COO((3, 3), [1, 2, 3], [0, 1, 2], [0, 1, 2])
dense = coo.to_dense()
print(dense) # Output: [[1. 0. 0.]
#                    [0. 2. 0.]
#                    [0. 0. 3.]]

4.4.2 Compressed Sparse Row (CSR) Example

import numpy as np

class CSR:
   def __init__(self, shape, data, indices, indptr):
       self.shape = shape
       self.data = data
       self.indices = indices
       self.indptr = indptr

   def to_dense(self):
       dense = np.zeros(self.shape)
       for i in range(len(self.data)):
           dense[self.indices[i], self.indptr[i]:self.indptr[i+1]] = self.data[i]
       return dense

# Test CSR
csr = CSR((3, 3), [1, 2, 3], [0, 1, 2], [0, 2, 5])
dense = csr.to_dense()
print(dense) # Output: [[1. 0. 0.]
#                    [0. 2. 0.]
#                    [0. 0. 3.]]

实际应用场景

5.1 Redis Data Structures Applications

• Real-time analytics and monitoring: Redis can be used to store and process real-time data streams, such as logs, metrics, and events.
• Session management: Redis can be used to store and manage user sessions in web applications, providing fast access to session data and reducing the load on databases.
• Caching: Redis can be used as a cache to improve the performance of applications by storing frequently accessed data in memory.
• Leaderboards and counters: Redis can be used to implement leaderboards and counters for online games and social media platforms, providing fast and scalable solutions for ranking and aggregating data.

5.2 Embeddings Applications

• Natural language processing: Word embeddings are widely used in natural language processing tasks, such as text classification, sentiment analysis, machine translation, and question answering.
• Computer vision: Image embeddings are used in computer vision tasks, such as image recognition, object detection, and segmentation.
• Recommendation systems: User and item embeddings are used in recommendation systems to learn latent representations of users and items, and to generate personalized recommendations based on similarity or collaborative filtering.
• Fraud detection: User and transaction embeddings are used in fraud detection to identify patterns and anomalies in user behavior and transactions, and to detect potential fraudulent activities.

5.3 Sparse Features Applications

• Network traffic analysis: Count Sketch and Bloom Filter can be used to analyze network traffic data, such as packets and flows, and to detect suspicious activities, such as DDoS attacks and malware.
• Text compression: Count Sketch can be used to compress large text corpora, such as books and documents, and to reduce the storage space and transmission time.
• Ad targeting: Bloom Filter can be used to target ads to specific audiences based on their interests, behaviors, and demographics, and to avoid showing irrelevant or annoying ads.

5.4 Sparse Tensors Applications

• Deep learning: Sparse tensors are used in deep learning models, such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs), to represent high-dimensional data with low memory footprint and computational cost.
• Graph algorithms: Sparse matrices and tensors are used in graph algorithms, such as PageRank, community detection, and graph traversal, to represent and manipulate large graphs efficiently.
• Scientific computing: Sparse matrices and tensors are used in scientific computing, such as linear algebra, numerical analysis, and optimization, to solve complex problems with large and sparse data structures.

工具和资源推荐

6.1 Redis Data Structures Tools

• RedisInsight: A graphical user interface for managing and monitoring Redis instances, providing visualization, debugging, and profiling tools.
• redis-cli: A command-line interface for interacting with Redis instances, providing commands for data manipulation, configuration, and diagnostics.
• Redis Commander: A web-based user interface for managing and monitoring Redis instances, providing a simple and intuitive interface for data exploration, querying, and editing.

6.2 Embeddings Tools

• Gensim: A Python library for topic modeling, document indexing, and similarity analysis, providing implementations of popular word embedding algorithms, such as Word2Vec and FastText.
• Spacy: A Python library for natural language processing, providing pre-trained word embeddings and transformer models for various NLP tasks, such as named entity recognition, part-of-speech tagging, and dependency parsing.
• TensorFlow: A machine learning framework for building and training deep learning models, providing support for various types of embeddings, such as word, image, and graph embeddings.

6.3 Sparse Features Tools

• Scikit-learn: A machine learning library for Python, providing implementations of probabilistic data structures, such as Count Sketch and Bloom Filter, and algorithms for feature selection, dimensionality reduction, and clustering.
• Faiss: A library for efficient similarity search and clustering of dense vectors, providing support for various distance metrics and indexing techniques, and interfaces for Python, C++, and Java.
• Annoy: A library for approximate nearest neighbor search in high-dimensional spaces, providing support for various distance metrics and indexing techniques, and interfaces for Python and C++.

6.4 Sparse Tensors Tools

• TensorFlow: A machine learning framework for building and training deep learning models, providing support for sparse tensors and operations, such as matrix multiplication, convolution, and pooling.
• PyTorch: A machine learning framework for building and training deep learning models, providing support for sparse tensors and operations, such as matrix multiplication, convolution, and pooling.
• SciPy: A scientific computing library for Python, providing support for sparse matrices and tensors, and algorithms for linear algebra, numerical analysis, and optimization.

总结：未来发展趋势与挑战

7.1 Redis Data Structures Future Directions

• Real-time analytics and streaming: Improving the performance and scalability of real-time analytics and streaming applications, such as logs, metrics, and events, by optimizing data structures, algorithms, and protocols.
• In-memory databases: Enhancing the functionality and performance of in-memory databases, such as Redis, by adding new data types, indexes, and query languages, and by integrating with other data sources and services.
• Cloud-native databases: Building cloud-native databases, such as Redis on AWS, Azure, and Google Cloud, by leveraging cloud infrastructure, such as containers, Kubernetes, and serverless computing.

7.2 Embeddings Future Directions

• Transfer learning: Developing transfer learning methods and models for embeddings, such as fine-tuning and multitask learning, to improve the generalization and adaptability of embeddings to different domains and tasks.
• Multimodal embeddings: Developing multimodal embeddings that can represent multiple modalities, such as text, images, audio, and video, and learn joint representations that capture the correlations and interactions between modalities.
• Explainable embeddings: Developing explainable embeddings that can provide insights and interpretations of the learned representations, and help users understand the underlying patterns, structures, and relationships in the data.

7.3 Sparse Features Future Directions

• Distributed probabilistic data structures: Developing distributed probabilistic data structures, such as Count Sketch and Bloom Filter, that can scale to large clusters and handle high-throughput and low-latency workloads, and provide fault tolerance, consistency, and availability.
• Adaptive feature selection: Developing adaptive feature selection methods and models that can dynamically select relevant features based on the input data, context, and objectives, and optimize the trade-off between accuracy and complexity.
• Online learning and adaptation: Developing online learning and adaptation methods and models that can continuously learn from streaming data, update the models in real-time, and adapt to changes and drifts in the data distribution.

7.4 Sparse Tensors Future Directions

• Quantized and compressed sparse tensors: Developing quantized and compressed sparse tensors that can reduce the memory footprint and computational cost of deep learning models, and enable efficient storage and transmission of large and sparse data structures.
• Automated sparsification and pruning: Developing automated sparsification and pruning methods and models that can identify and remove redundant or irrelevant connections in deep learning models, and optimize the network architecture and topology.
• Distributed and parallel sparse tensor computation: Developing distributed and parallel sparse tensor computation methods and frameworks that can scale to large clusters and handle high-throughput and low-latency workloads, and provide fault tolerance, consistency, and availability.

附录：常见问题与解答

8.1 Redis Data Structures FAQ

Q: What is the difference between String and Hash?

A: String is a simple key-value pair, where the value is a string. Hash is a complex key-value pair, where the value is a hash table with multiple fields and values. String is suitable for storing small and simple data, while Hash is suitable for storing large and complex data.

Q: How to store an array in Redis?

A: There are several ways to store an array in Redis, depending on the type and structure of the array. For example, you can use a List for a simple array of strings, or a Hash for a structured array of key-value pairs, or a Set for a unique array of elements, or a Sorted Set for a ranked array of elements.

Q: How to implement a counter in Redis?

A: You can implement a counter in Redis using the INCR command, which increments the value of a key by one, or the INCRBY command, which increments the value of a key by a specified amount. You can also use the DECR command to decrement the value of a key by one, or the DECRBY command to decrement the value of a key by a specified amount.

8.2 Embeddings FAQ

Q: How to choose the dimension of word embeddings?

A: The dimension of word embeddings depends on the size and complexity of the corpus, the model architecture, and the task requirements. A smaller dimension may result in underfitting, while a larger dimension may result in overfitting. A common practice is to set the dimension to a few hundred, such as 100 or 300, based on empirical evidence and experience.

Q: How to initialize word embeddings?

A: You can initialize word embeddings using various strategies, such as random initialization, zero initialization, pre-trained initialization, or hybrid initialization. Random initialization generates random vectors for each word, while zero initialization sets all vectors to zeros. Pre-trained initialization uses pre-trained vectors from external resources, such as Word2Vec or GloVe, while hybrid initialization combines random or zero initialization with pre-trained initialization.

Q: How to evaluate word embeddings?

A: You can evaluate word embeddings using various metrics, such as intrinsic evaluation metrics, extrinsic evaluation metrics, or qualitative evaluation metrics. Intrinsic evaluation metrics measure the quality and coherence of the embeddings themselves, such as similarity, clustering, or visualization. Extrinsic evaluation metrics measure the performance and effectiveness of the embeddings in downstream tasks, such as classification, regression, or generation. Qualitative evaluation metrics involve human judgment and feedback, such as analogy tests, error analysis, or user studies.

8.3 Sparse Features FAQ

Q: How to select the number of hash functions in Count Sketch?

A: The number of hash functions in Count Sketch depends on the desired accuracy and confidence level of the estimates, and the size and sparsity of the data. A larger number of hash functions may increase the accuracy and confidence level of the estimates, but may also increase the space and time complexity of the sketch. A common practice is to set the number of hash functions to a few times the expected number of non-zero elements in the data, based on empirical evidence and experience.

Q: How to select the size of Bloom Filter?

A: The size of Bloom Filter depends on the expected number of elements, the desired false positive rate, and the size and sparsity of the data. A larger size may decrease the false positive rate, but may also increase the space and time complexity of the filter. A common practice is to set the size to a few times the expected number of elements, based on empirical evidence and experience, and adjust the false positive rate accordingly.

Q: How to compress sparse matrices and tensors?

A: You can compress sparse matrices and tensors using various techniques, such as coordinate format (COO), compressed sparse row (CSR), or block sparse row (BSR). These techniques represent the non-zero elements of the matrix or tensor using coordinates and values, and eliminate the redundant zeros, thus reducing the memory footprint and computational cost. You can also apply lossy compression techniques, such as quantization or approximation, to further reduce the storage size and transmission time.

8.4 Sparse Tensors FAQ

Q: How to convert dense tensors to sparse tensors?

A: You can convert dense tensors to sparse tensors using various methods, such as thresholding, sampling, or hashing. Thresholding converts all elements below a certain threshold to zeros, and keeps only the non-zero elements as the sparse tensor. Sampling randomly selects a subset of elements from the dense tensor, and creates a sparse tensor from the selected elements. Hashing maps the dense tensor to a sparse hash table, where each bucket contains a list of non-zero elements with the same hash value.

Q: How to perform sparse matrix multiplication?

A: You can perform sparse matrix multiplication using various algorithms, such as the coordinate format (COO) algorithm, the compressed sparse row (CSR) algorithm, or the blocked coordinate format (BCF) algorithm. These algorithms represent the sparse matrices using efficient data structures, such as COO or CSR, and perform the matrix multiplication using optimized operations, such as sparse-dense or sparse-sparse multiplication.

Q: How to implement sparse convolutional neural networks?

A: You can implement sparse convolutional neural networks using various frameworks, such as TensorFlow, PyTorch, or MXNet. These frameworks provide built-in support for sparse tensors and operations, such as sparse-dense multiplication, sparse-sparse multiplication, and sparse convolution. You can also use third-party libraries, such as CuSparse, MKL, or Eigen, to implement custom kernels and operators for sparse tensors.