{"id":63445,"date":"2025-04-16T13:13:18","date_gmt":"2025-04-16T13:13:18","guid":{"rendered":""},"modified":"2025-04-16T13:13:18","modified_gmt":"2025-04-16T13:13:18","slug":"%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-63445","status":"publish","type":"product","link":"https:\/\/express24.ir\/d\/product\/%d8%aa%d8%b1%d8%ac%d9%85%d9%87-%d9%81%d8%a7%d8%b1%d8%b3%db%8c-%d9%85%d9%82%d8%a7%d9%84%d9%87-63445\/","title":{"rendered":"\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0628\u0627 \u0627\u0628\u0639\u0627\u062f \u0628\u0627\u0644\u0627 \u0628\u0631\u0627\u06cc PCA \u062a\u0627\u0646\u0633\u0648\u0631 \u0686\u0646\u062f \u0636\u0644\u0639\u06cc"},"content":{"rendered":"\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nHigh-dimensional optimization for multi-spiked tensor PCA<\/td>\n<\/tr>\n
\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0628\u0627 \u0627\u0628\u0639\u0627\u062f \u0628\u0627\u0644\u0627 \u0628\u0631\u0627\u06cc PCA \u062a\u0627\u0646\u0633\u0648\u0631 \u0686\u0646\u062f \u0636\u0644\u0639\u06cc<\/td>\n<\/tr>\n
\u0646\u0648\u06cc\u0633\u0646\u062f\u06af\u0627\u0646 <\/td>\nG\u00e9rard Ben Arous, C\u00e9dric Gerbelot, Vanessa Piccolo<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc <\/td>\nPDF<\/td>\n<\/tr>\n
\u0632\u0628\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u062a\u062d\u0648\u06cc\u0644\u06cc <\/td>\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc<\/td>\n<\/tr>\n
\u0641\u0631\u0645\u062a \u0645\u0642\u0627\u0644\u0647 \u062a\u0631\u062c\u0645\u0647 \u0634\u062f\u0647 <\/td>\n\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\n<\/tr>\n
\u0646\u062d\u0648\u0647 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 <\/td>\n\u062f\u0648 \u062a\u0627 \u0633\u0647 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634 (\u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u062f\u0627\u0646\u0644\u0648\u062f\u06cc)<\/td>\n<\/tr>\n
\u062a\u0639\u062f\u0627\u062f \u0635\u0641\u062d\u0627\u062a<\/td>\n126<\/td>\n<\/tr>\n
\u0644\u06cc\u0646\u06a9 \u062f\u0627\u0646\u0644\u0648\u062f \u0631\u0627\u06cc\u06af\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc<\/td>\n\u062f\u0627\u0646\u0644\u0648\u062f \u0645\u0642\u0627\u0644\u0647<\/a><\/td>\n<\/tr>\n
\u062f\u0633\u062a\u0647 \u0628\u0646\u062f\u06cc \u0645\u0648\u0636\u0648\u0639\u0627\u062a <\/td>\nMachine Learning,Machine Learning,Probability,Statistics Theory,\u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 , \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 , \u0627\u062d\u062a\u0645\u0627\u0644 , \u062a\u0626\u0648\u0631\u06cc \u0622\u0645\u0627\u0631 ,<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a <\/td>\nSubmitted 12 August, 2024; originally announced August 2024. , MSC Class: 68Q87; 62F10; 62F30; 62M05<\/td>\n<\/tr>\n
\u062a\u0648\u0636\u06cc\u062d\u0627\u062a \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc <\/td>\n\u0627\u0631\u0627\u0626\u0647 \u0634\u062f\u0647 12 \u0627\u0648\u062a 2024 \u061b\u062f\u0631 \u0627\u0628\u062a\u062f\u0627 \u0627\u0648\u062a 2024 \u0627\u0639\u0644\u0627\u0645 \u0634\u062f. \u060c \u06a9\u0644\u0627\u0633 MSC: 68Q87 \u061b62F10 \u061b62F30 \u061b62M05<\/td>\n<\/tr>\n
\u0627\u0637\u0644\u0627\u0639\u0627\u062a \u0628\u06cc\u0634\u062a\u0631 \u0627\u0632 \u0627\u06cc\u0646 \u0645\u0642\u0627\u0644\u0647 \u062f\u0631 \u067e\u0627\u06cc\u06af\u0627\u0647 \u0647\u0627\u06cc \u0639\u0644\u0645\u06cc <\/td>\n\n INSPIRE HEP<\/a>
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\n NASA ADS<\/a>
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\n Google Scholar<\/a>
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\n Semantic Scholar<\/a>
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\n arXiv<\/a><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n

\u0686\u06a9\u06cc\u062f\u0647<\/h2>\n

We study the dynamics of two local optimization algorithms, online stochastic gradient descent (SGD) and gradient flow, within the framework of the multi-spiked tensor model in the high-dimensional regime. This multi-index model arises from the tensor principal component analysis (PCA) problem, which aims to infer $r$ unknown, orthogonal signal vectors within the $N$-dimensional unit sphere through maximum likelihood estimation from noisy observations of an order-$p$ tensor. We determine the number of samples and the conditions on the signal-to-noise ratios (SNRs) required to efficiently recover the unknown spikes from natural initializations. Specifically, we distinguish between three types of recovery: exact recovery of each spike, recovery of a permutation of all spikes, and recovery of the correct subspace spanned by the signal vectors. We show that with online SGD, it is possible to recover all spikes provided a number of sample scaling as $N^{p-2}$, aligning with the computational threshold identified in the rank-one tensor PCA problem [Ben Arous, Gheissari, Jagannath 2020, 2021]. For gradient flow, we show that the algorithmic threshold to efficiently recover the first spike is also of order $N^{p-2}$. However, recovering the subsequent directions requires the number of samples to scale as $N^{p-1}$. Our results are obtained through a detailed analysis of a low-dimensional system that describes the evolution of the correlations between the estimators and the spikes. In particular, the hidden vectors are recovered one by one according to a sequential elimination phenomenon: as one correlation exceeds a critical threshold, all correlations sharing a row or column index decrease and become negligible, allowing the subsequent correlation to grow and become macroscopic. The sequence in which correlations become macroscopic depends on their initial values and on the associated SNRs.<\/p>\n

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\u0646\u0648\u06cc\u0632 (SNR) \u0645\u0648\u0631\u062f \u0646\u06cc\u0627\u0632 \u0628\u0631\u0627\u06cc \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u06a9\u0627\u0631\u0622\u0645\u062f \u0633\u0646\u0628\u0644\u0647 \u0647\u0627\u06cc \u0646\u0627\u0634\u0646\u0627\u062e\u062a\u0647 \u0627\u0632 \u0627\u0648\u0644\u06cc\u0647 \u0633\u0627\u0632\u06cc \u0637\u0628\u06cc\u0639\u06cc \u0631\u0627 \u062a\u0639\u06cc\u06cc\u0646 \u0645\u06cc \u06a9\u0646\u06cc\u0645.\u0628\u0647 \u0637\u0648\u0631 \u062e\u0627\u0635 \u060c \u0645\u0627 \u0628\u06cc\u0646 \u0633\u0647 \u0646\u0648\u0639 \u0628\u0647\u0628\u0648\u062f\u06cc \u062a\u0645\u0627\u06cc\u0632 \u0642\u0627\u0626\u0644 \u0645\u06cc \u0634\u0648\u06cc\u0645: \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u062f\u0642\u06cc\u0642 \u0647\u0631 \u0633\u0646\u0628\u0644\u0647 \u060c \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u062c\u0627\u06cc\u06af\u0634\u062a \u0647\u0645\u0647 \u0633\u0646\u0628\u0644\u0647 \u0647\u0627 \u0648 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\u062f\u0631 \u0645\u0634\u06a9\u0644 Tensor PCA \u0631\u062a\u0628\u0647 \u0628\u0646\u062f\u06cc \u0645\u06cc \u0634\u0648\u062f [\u0628\u0646 \u0628\u0631\u0627\u0646\u06af\u06cc\u062e\u062a\u06af\u06cc \u060c \u0642\u0633\u0631\u06cc\u0633\u0627\u0631\u06cc\u060c \u062c\u0627\u06af\u0627\u0646\u0627\u062a 2020 \u060c 2021].\u0628\u0631\u0627\u06cc \u062c\u0631\u06cc\u0627\u0646 \u0634\u06cc\u0628 \u060c \u0645\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u06a9\u0647 \u0622\u0633\u062a\u0627\u0646\u0647 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645\u06cc \u0628\u0631\u0627\u06cc \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u06a9\u0627\u0631\u0622\u0645\u062f \u0627\u0648\u0644\u06cc\u0646 \u0633\u0646\u0628\u0644\u0647 \u0646\u06cc\u0632 \u0627\u0632 \u0633\u0641\u0627\u0631\u0634 $ n^{p-2} $ \u0627\u0633\u062a.\u0628\u0627 \u0627\u06cc\u0646 \u062d\u0627\u0644 \u060c \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u062f\u0633\u062a\u0648\u0631\u0627\u0644\u0639\u0645\u0644 \u0647\u0627\u06cc \u0628\u0639\u062f\u06cc \u0628\u0647 \u062a\u0639\u062f\u0627\u062f \u0646\u0645\u0648\u0646\u0647 \u0647\u0627 \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u062f \u062a\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 $ n^{p-1} $ \u0645\u0642\u06cc\u0627\u0633 \u0634\u0648\u062f.\u0646\u062a\u0627\u06cc\u062c \u0645\u0627 \u0627\u0632 \u0637\u0631\u06cc\u0642 \u062a\u062c\u0632\u06cc\u0647 \u0648 \u062a\u062d\u0644\u06cc\u0644 \u062f\u0642\u06cc\u0642 \u0627\u0632 \u06cc\u06a9 \u0633\u06cc\u0633\u062a\u0645 \u06a9\u0645 \u0628\u0639\u062f\u06cc \u06a9\u0647 \u062a\u0648\u0635\u06cc\u0641 \u062a\u06a9\u0627\u0645\u0644 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0628\u06cc\u0646 \u0628\u0631\u0622\u0648\u0631\u062f\u06af\u0631\u0647\u0627 \u0648 \u0633\u0646\u0628\u0644\u0647 \u0647\u0627 \u0627\u0633\u062a \u060c \u0628\u062f\u0633\u062a \u0645\u06cc \u0622\u06cc\u062f.\u0628\u0647 \u0637\u0648\u0631 \u062e\u0627\u0635 \u060c \u0628\u0631\u062f\u0627\u0631\u0647\u0627\u06cc \u067e\u0646\u0647\u0627\u0646 \u06cc\u06a9 \u0628\u0647 \u06cc\u06a9 \u0645\u0637\u0627\u0628\u0642 \u0628\u0627 \u06cc\u06a9 \u067e\u062f\u06cc\u062f\u0647 \u0627\u0632 \u0628\u06cc\u0646 \u0628\u0631\u062f\u0646 \u067e\u06cc \u062f\u0631 \u067e\u06cc \u0628\u0627\u0632\u06cc\u0627\u0628\u06cc \u0645\u06cc \u0634\u0648\u0646\u062f: \u0627\u0632 \u0622\u0646\u062c\u0627 \u06a9\u0647 \u06cc\u06a9 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0627\u0632 \u06cc\u06a9 \u0622\u0633\u062a\u0627\u0646\u0647 \u0628\u062d\u0631\u0627\u0646\u06cc \u0641\u0631\u0627\u062a\u0631 \u0645\u06cc \u0631\u0648\u062f \u060c \u062a\u0645\u0627\u0645 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0647\u0627\u06cc\u06cc \u06a9\u0647 \u062f\u0627\u0631\u0627\u06cc \u06cc\u06a9 \u0631\u062f\u06cc\u0641 \u06cc\u0627 \u0634\u0627\u062e\u0635 \u0633\u062a\u0648\u0646 \u0647\u0633\u062a\u0646\u062f \u06a9\u0627\u0647\u0634 \u0645\u06cc \u06cc\u0627\u0628\u062f \u0648 \u0646\u0627\u0686\u06cc\u0632 \u0645\u06cc \u0634\u0648\u062f \u0648 \u0628\u0627\u0639\u062b \u0645\u06cc \u0634\u0648\u062f \u06a9\u0647 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0628\u0639\u062f\u06cc \u0631\u0634\u062f \u06a9\u0646\u062f \u0648 \u0645\u0627\u06a9\u0631\u0648\u0633\u06a9\u0648\u067e\u06cc \u0634\u0648\u062f.\u062f\u0646\u0628\u0627\u0644\u0647 \u0627\u06cc \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u0647\u0645\u0628\u0633\u062a\u06af\u06cc \u0645\u06cc \u0634\u0648\u062f \u0645\u0627\u06a9\u0631\u0648\u0633\u06a9\u0648\u067e\u06cc \u0628\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0627\u0648\u0644\u06cc\u0647 \u0622\u0646\u0647\u0627 \u0648 SNR \u0647\u0627\u06cc \u0645\u0631\u062a\u0628\u0637 \u0628\u0633\u062a\u06af\u06cc \u062f\u0627\u0631\u062f.<\/p>\n\r\n\r\n \r\n \r\n \r\n \r\n
\u0641\u0631\u0645\u062a \u0627\u0631\u0627\u0626\u0647 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u062a\u062d\u0648\u06cc\u0644 \u0628\u0647 \u0635\u0648\u0631\u062a \u0641\u0627\u06cc\u0644 \u0648\u0631\u062f<\/td>\r\n <\/tr>\r\n
\u0632\u0645\u0627\u0646 \u062a\u062d\u0648\u06cc\u0644 \u062a\u0631\u062c\u0645\u0647 \u0645\u0642\u0627\u0644\u0647 <\/td>\r\n \u0628\u06cc\u0646 2 \u062a\u0627 3 \u0631\u0648\u0632 \u067e\u0633 \u0627\u0632 \u062b\u0628\u062a \u0633\u0641\u0627\u0631\u0634<\/td>\r\n <\/tr>\r\n\t
\u06a9\u06cc\u0641\u06cc\u062a \u062a\u0631\u062c\u0645\u0647 <\/td>\r\n \u0628\u0633\u06cc\u0627\u0631 \u0628\u0627\u0644\u0627. \u0645\u0642\u0627\u0644\u0647 \u0641\u0642\u0637 \u062a\u0648\u0633\u0637 \u0645\u062a\u0631\u062c\u0645\u06cc\u0646 \u0628\u0627 \u0645\u062f\u0631\u06a9 \u062f\u0627\u0646\u0634\u06af\u0627\u0647\u06cc \u0645\u062a\u0631\u062c\u0645\u06cc \u062a\u0631\u062c\u0645\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f.<\/td>\r\n <\/tr>\r\n\t\t
\u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 <\/td>\r\n \u06a9\u0644\u06cc\u0647 \u062c\u062f\u0627\u0648\u0644 \u0648 \u0641\u0631\u0645\u0648\u0644 \u0647\u0627 \u0646\u06cc\u0632 \u062f\u0631 \u0641\u0627\u06cc\u0644 \u062a\u062d\u0648\u06cc\u0644\u06cc \u0648\u0631\u062f \u062f\u0631\u062c \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/td>\r\n <\/tr>\r\n<\/table>\r\n\r\n\n","protected":false},"excerpt":{"rendered":"

\u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0627\u0646\u06af\u0644\u06cc\u0633\u06cc High-dimensional optimization for multi-spiked tensor PCA \u0639\u0646\u0648\u0627\u0646 \u0645\u0642\u0627\u0644\u0647 \u0628\u0647 \u0641\u0627\u0631\u0633\u06cc \u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0628\u0627 […]<\/p>\n","protected":false},"featured_media":27,"comment_status":"open","ping_status":"open","template":"","meta":{"pmpro_default_level":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-gradient":""}}},"product_cat":[21],"product_tag":[],"class_list":{"0":"post-63445","1":"product","2":"type-product","3":"status-publish","4":"has-post-thumbnail","6":"product_cat-21","7":"pmpro-has-access","8":"desktop-align-left","9":"tablet-align-left","10":"mobile-align-left","12":"first","13":"instock","14":"downloadable","15":"shipping-taxable","16":"purchasable","17":"product-type-simple"},"yoast_head":"\n\u062a\u0631\u062c\u0645\u0647 \u0641\u0627\u0631\u0633\u06cc \u0645\u0642\u0627\u0644\u0647 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0628\u0627 \u0627\u0628\u0639\u0627\u062f \u0628\u0627\u0644\u0627 \u0628\u0631\u0627\u06cc PCA \u062a\u0627\u0646\u0633\u0648\u0631 \u0686\u0646\u062f \u0636\u0644\u0639\u06cc - 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