如何使用SVD和TensorFlow核心API压缩图像

内容概览 介绍 设置 基本SVD 低级接近与SVD 图像加载 图像压缩算法 评估近似性 压缩因子 VS 等级 单一值的累积总和 错误和单一值 结论 介绍 本教程探讨了技术的 (SVD) 和其应用于低级接近问题. SVD 用于因素化真实或复杂的矩阵,并且在数据科学中有各种用例,如图像压缩。 项目。 单一价值分解 图像 设置 import matplotlib from matplotlib.image import imread from matplotlib import pyplot as plt import requests # Preset Matplotlib figure sizes. matplotlib.rcParams['figure.figsize'] = [16, 9] import tensorflow as tf print(tf.__version__) 2024-08-15 02:40:31.007111: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:485] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered 2024-08-15 02:40:31.028202: E external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:8454] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered 2024-08-15 02:40:31.034794: E external/local_xla/xla/stream_executor/cuda/cuda_blas.cc:1452] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered 2.17.0 基本SVD A 矩阵的单值分解是由以下因素化决定的: A = 已知 哪里 Am×n:输入矩阵,其中 m≥n Um×n: 正方形矩阵, UTU=I,每个列, ui,表示A的左单个向量 Σn×n:每一个圆形输入的圆形矩阵, σi,表示A的单数值 VTn×n: 正方形矩阵, VTV=I,每个行, vi,表示 A 的右单向矢量 当m<n时,U和 Σ都有尺寸(m×m),而VT有尺寸(m×n)。 TensorFlow 的线性 algebra 包具有一个函数, ,可用来计算一个或多个矩阵的单值分解,从定义一个简单的矩阵开始,并计算其SVD因素化。 tf.linalg.svd A = tf.random.uniform(shape=[40,30]) # Compute the SVD factorization s, U, V = tf.linalg.svd(A) # Define Sigma and V Transpose S = tf.linalg.diag(s) V_T = tf.transpose(V) # Reconstruct the original matrix A_svd = U@S@V_T # Visualize plt.bar(range(len(s)), s); plt.xlabel("Singular value rank") plt.ylabel("Singular value") plt.title("Bar graph of singular values"); WARNING: All log messages before absl::InitializeLog() is called are written to STDERR I0000 00:00:1723689633.424919 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.428762 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.432420 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.437707 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.449819 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.453529 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.457079 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.460600 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.464019 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.467451 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.470857 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689633.474376 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.696864 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.698995 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.701043 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.703182 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.705235 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.707223 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.709163 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.711162 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.713114 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.715090 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.717036 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.719041 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.757916 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.759983 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.761969 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.764105 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.766234 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.768229 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.770180 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.772176 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.774158 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.776614 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.779022 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1723689634.781451 126589 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 该 函数可以直接从输出中计算矩阵重构。 . tf.einsum tf.linalg.svd A_svd = tf.einsum('s,us,vs -> uv',s,U,V) print('\nReconstructed Matrix, A_svd', A_svd) Reconstructed Matrix, A_svd tf.Tensor( [[0.819255 0.95082176 0.30085772 ... 0.34306473 0.9112067 0.45277762] [0.71988434 0.20786765 0.3098724 ... 0.82478774 0.82917416 0.19994356] [0.23278469 0.17072585 0.10488278 ... 0.60508984 0.8121238 0.10594818] ... [0.10953172 0.72077066 0.30288345 ... 0.5790409 0.15389094 0.11265922] [0.44894537 0.04313891 0.6947845 ... 0.41589245 0.6668094 0.53494996] [0.79140556 0.47814542 0.24980234 ... 0.32851136 0.86007696 0.32692394]], shape=(40, 30), dtype=float32) 与SVD的低级接近 矩阵A的等级由其列所覆盖的矢量空间的尺寸决定,SVD可以用来接近较低等级的矩阵,最终减少存储矩阵所代表的信息所需的数据的维度。 A 的 SVD 的排名-r 接近是通过公式来定义的: Ar=UrΣrVrT 哪里 Urm×r:由U的第一个r列组成的矩阵 Σrr×r:由 Σ 中的第一个 r 单数值组成的直径矩阵 Vrr×nT:由VT的第一个r行组成的矩阵 开始写一个函数来计算给定矩阵的Rang-r 接近度 这个低级接近度程序用于图像压缩;因此,也很有用来计算每个接近度的物理数据大小。为了简单,假设一个Rang-r 接近度矩阵的数据大小等于计算接近度所需的元素总数。接下来,写一个函数来可视化原始矩阵,A 的Rang-r 接近度,Ar 和错误矩阵,A−Ar 。 def rank_r_approx(s, U, V, r, verbose=False): # Compute the matrices necessary for a rank-r approximation s_r, U_r, V_r = s[..., :r], U[..., :, :r], V[..., :, :r] # ... implies any number of extra batch axes # Compute the low-rank approximation and its size A_r = tf.einsum('...s,...us,...vs->...uv',s_r,U_r,V_r) A_r_size = tf.size(U_r) + tf.size(s_r) + tf.size(V_r) if verbose: print(f"Approximation Size: {A_r_size}") return A_r, A_r_size def viz_approx(A, A_r): # Plot A, A_r, and A - A_r vmin, vmax = 0, tf.reduce_max(A) fig, ax = plt.subplots(1,3) mats = [A, A_r, abs(A - A_r)] titles = ['Original A', 'Approximated A_r', 'Error |A - A_r|'] for i, (mat, title) in enumerate(zip(mats, titles)): ax[i].pcolormesh(mat, vmin=vmin, vmax=vmax) ax[i].set_title(title) ax[i].axis('off') print(f"Original Size of A: {tf.size(A)}") s, U, V = tf.linalg.svd(A) Original Size of A: 1200 # Rank-15 approximation A_15, A_15_size = rank_r_approx(s, U, V, 15, verbose = True) viz_approx(A, A_15) Approximation Size: 1065 # Rank-3 approximation A_3, A_3_size = rank_r_approx(s, U, V, 3, verbose = True) viz_approx(A, A_3) Approximation Size: 213 正如预期的那样,使用较低的等级会导致不太准确的接近。然而,这些较低的接近的质量在现实世界的场景中往往足够好。 另外,请注意,使用SVD的低级接近的主要目标是减少数据的尺寸性,而不是减少数据本身的磁盘空间。 图像加载 以下图像可在 图像是一个由谷歌研究的大脑团队开发的文本到图像传播模型,人工智能基于提示创建了这个图像:“在泰晤士广场骑自行车的科吉狗的照片。它戴着太阳镜和海滩帽子。 这是多么酷! 您也可以更改下面的URL到任何 .jpg 链接来加载自定义图像。 图像 在阅读JPEG文件后,Matplotlib输出一个矩阵,I,的形状(m×n×3)代表一个二维图像,分别为红色,绿色和蓝色的3种颜色频道。 img_link = "https://imagen.research.google/main_gallery_images/a-photo-of-a-corgi-dog-riding-a-bike-in-times-square.jpg" img_path = requests.get(img_link, stream=True).raw I = imread(img_path, 0) print("Input Image Shape:", I.shape) Input Image Shape: (1024, 1024, 3) def show_img(I): # Display the image in matplotlib img = plt.imshow(I) plt.axis('off') return show_img(I) 图像压缩算法 现在,使用SVD来计算样本图像的低级接近性。请记住,图像是形状的(1024×1024×3),并且SVD理论仅适用于二维矩阵。这意味着样本图像必须分组成3个相等大小的矩阵,这对应于3种颜色渠道中的每一个。这可以通过将矩阵转换为形状(3×1024×1024)来完成。为了清楚地可视化接近错误,请将图像的RGB值从(0,255)转移到(0,1)。 这个功能是有用的。 tf.clip_by_value def compress_image(I, r, verbose=False): # Compress an image with the SVD given a rank I_size = tf.size(I) print(f"Original size of image: {I_size}") # Compute SVD of image I = tf.convert_to_tensor(I)/255 I_batched = tf.transpose(I, [2, 0, 1]) # einops.rearrange(I, 'h w c -> c h w') s, U, V = tf.linalg.svd(I_batched) # Compute low-rank approximation of image across each RGB channel I_r, I_r_size = rank_r_approx(s, U, V, r) I_r = tf.transpose(I_r, [1, 2, 0]) # einops.rearrange(I_r, 'c h w -> h w c') I_r_prop = (I_r_size / I_size) if verbose: # Display compressed image and attributes print(f"Number of singular values used in compression: {r}") print(f"Compressed image size: {I_r_size}") print(f"Proportion of original size: {I_r_prop:.3f}") ax_1 = plt.subplot(1,2,1) show_img(tf.clip_by_value(I_r,0.,1.)) ax_1.set_title("Approximated image") ax_2 = plt.subplot(1,2,2) show_img(tf.clip_by_value(0.5+abs(I-I_r),0.,1.)) ax_2.set_title("Error") return I_r, I_r_prop 现在,计算以下等级的Rang-r 接近: 100, 50, 10 I_100, I_100_prop = compress_image(I, 100, verbose=True) Original size of image: 3145728 Number of singular values used in compression: 100 Compressed image size: 614700 Proportion of original size: 0.195 I_50, I_50_prop = compress_image(I, 50, verbose=True) Original size of image: 3145728 Number of singular values used in compression: 50 Compressed image size: 307350 Proportion of original size: 0.098 I_10, I_10_prop = compress_image(I, 10, verbose=True) Original size of image: 3145728 Number of singular values used in compression: 10 Compressed image size: 61470 Proportion of original size: 0.020 评估近似性 有各种有趣的方法来衡量效率,并对矩阵接近进行更多的控制。 压缩因子 VS 等级 对于上述每一个接近,观察数据大小如何随着排名而变化。 plt.figure(figsize=(11,6)) plt.plot([100, 50, 10], [I_100_prop, I_50_prop, I_10_prop]) plt.xlabel("Rank") plt.ylabel("Proportion of original image size") plt.title("Compression factor vs rank"); 要进一步探索这一点,请记住,一个接近矩阵的数据大小,Ar,是定义为其计算所需的元素总数。 x=(m×r)+r+(r×n)=r×(m+n+1) c=xy=r×(m+n+1)m×n 哪里 x:空气的大小 y : A 的尺寸 C = XY:压缩因子 r:接近的等级 m 和 n: A 的行和列维度 为了找到需要将图像压缩到所需因子的等级,r,c,上面的方程式可以重新安排以解决r: r = c×m×nm+n+1 请注意,这个公式是独立于颜色频道的尺寸,因为RGB的每一个接近不会相互影响,现在,写一个函数来压缩输入图像,根据所需的压缩因子。 def compress_image_with_factor(I, compression_factor, verbose=False): # Returns a compressed image based on a desired compression factor m,n,o = I.shape r = int((compression_factor * m * n)/(m + n + 1)) I_r, I_r_prop = compress_image(I, r, verbose=verbose) return I_r 将图像压缩到其原始大小的15%。 compression_factor = 0.15 I_r_img = compress_image_with_factor(I, compression_factor, verbose=True) Original size of image: 3145728 Number of singular values used in compression: 76 Compressed image size: 467172 Proportion of original size: 0.149 单一值的累积总和 单数值的累积总和可以是通过排名-r 接近捕获的能量的有用指标。 可视化样本图像中单数值的 RGB 平均累积比例。 这个功能可以有用。 tf.cumsum def viz_energy(I): # Visualize the energy captured based on rank # Computing SVD I = tf.convert_to_tensor(I)/255 I_batched = tf.transpose(I, [2, 0, 1]) s, U, V = tf.linalg.svd(I_batched) # Plotting average proportion across RGB channels props_rgb = tf.map_fn(lambda x: tf.cumsum(x)/tf.reduce_sum(x), s) props_rgb_mean = tf.reduce_mean(props_rgb, axis=0) plt.figure(figsize=(11,6)) plt.plot(range(len(I)), props_rgb_mean, color='k') plt.xlabel("Rank / singular value number") plt.ylabel("Cumulative proportion of singular values") plt.title("RGB-averaged proportion of energy captured by the first 'r' singular values") viz_energy(I) 看起来这个图像中的90%以上的能量被捕获在前100个单一值之内,现在,写一个函数来压缩输入图像,考虑到所需的能量保留因子。 def compress_image_with_energy(I, energy_factor, verbose=False): # Returns a compressed image based on a desired energy factor # Computing SVD I_rescaled = tf.convert_to_tensor(I)/255 I_batched = tf.transpose(I_rescaled, [2, 0, 1]) s, U, V = tf.linalg.svd(I_batched) # Extracting singular values props_rgb = tf.map_fn(lambda x: tf.cumsum(x)/tf.reduce_sum(x), s) props_rgb_mean = tf.reduce_mean(props_rgb, axis=0) # Find closest r that corresponds to the energy factor r = tf.argmin(tf.abs(props_rgb_mean - energy_factor)) + 1 actual_ef = props_rgb_mean[r] I_r, I_r_prop = compress_image(I, r, verbose=verbose) print(f"Proportion of energy captured by the first {r} singular values: {actual_ef:.3f}") return I_r 压缩图像以保留75%的能量。 energy_factor = 0.75 I_r_img = compress_image_with_energy(I, energy_factor, verbose=True) Original size of image: 3145728 Number of singular values used in compression: 35 Compressed image size: 215145 Proportion of original size: 0.068 Proportion of energy captured by the first 35 singular values: 0.753 错误和单一值 接近错误和单数值之间也存在着一个有趣的关系,结果显示,接近的方格弗罗比尼乌斯标准等于其单数值的方块的总和: 通过本教程开头的示例矩阵的排名10接近来测试这个关系。 s, U, V = tf.linalg.svd(A) A_10, A_10_size = rank_r_approx(s, U, V, 10) squared_norm = tf.norm(A - A_10)**2 s_squared_sum = tf.reduce_sum(s[10:]**2) print(f"Squared Frobenius norm: {squared_norm:.3f}") print(f"Sum of squared singular values left out: {s_squared_sum:.3f}") Squared Frobenius norm: 33.021 Sum of squared singular values left out: 33.021 结论 本笔记本介绍了使用 TensorFlow 实现单一值分解的过程,并将其应用于写成图像压缩算法。 TensorFlow Core API 可用于各种高性能科学计算用例。 要了解更多有关 TensorFlow 的线性 algebra 功能,请访问 linalg 模块的文档。 SVD也可以用于构建推荐系统。 有关使用 TensorFlow Core API 的更多示例,请参阅 如果您想了解有关数据加载和准备的更多信息,请参阅下面的教程 或 . 指南 图像数据加载 CSV 数据加载 本文最初在 TensorFlow 网站上发表,此文在新标题下出现在这里,并在 CC BY 4.0 版下获得许可。 本文最初在 TensorFlow 网站上发表,此文在新标题下出现在这里,并在 CC BY 4.0 版下获得许可。 腾讯 腾讯