{"url":"/method/holographic-reduced-representation","slug":"holographic-reduced-representation","name":"Holographic Reduced Representation","full_name":"Holographic Reduced Representation","full_name_withheld":false,"description_markdown":"**Holographic Reduced Representations** are a simple mechanism to represent an associative array of key-value pairs in a fixed-size vector. Each individual key-value pair is the same size as the entire associative array; the array is represented by the sum of the pairs. Concretely, consider a complex vector key $r = (a\\_{r}[1]e^{iφ\\_{r}[1]}, a\\_{r}[2]e^{iφ\\_{r}[2]}, \\dots)$, which is the same size as the complex vector value x. The pair is \"bound\" together by element-wise complex multiplication, which multiplies the moduli and adds the phases of the elements:\r\n\r\n$$ y = r \\otimes x $$\r\n\r\n$$ y =  \\left(a\\_{r}[1]a\\_{x}[1]e^{i(φ\\_{r}[1]+φ\\_{x}[1])}, a\\_{r}[2]a\\_{x}[2]e^{i(φ\\_{r}[2]+φ\\_{x}[2])}, \\dots\\right) $$\r\n\r\nGiven keys $r\\_{1}$, $r\\_{2}$, $r\\_{3}$ and input vectors $x\\_{1}$, $x\\_{2}$, $x\\_{3}$, the associative array is:\r\n\r\n$$c = r\\_{1} \\otimes x\\_{1} + r\\_{2} \\otimes x\\_{2} + r\\_{3} \\otimes x\\_{3} $$\r\n\r\nwhere we call $c$ a memory trace. Define the key inverse:\r\n\r\n$$ r^{-1} = \\left(a\\_{r}[1]^{−1}e^{−iφ\\_{r}[1]}, a\\_{r}[2]^{−1}e^{−iφ\\_{r}[2]}, \\dots\\right) $$\r\n\r\nTo retrieve the item associated with key $r\\_{k}$, we multiply the memory trace element-wise by the vector $r^{-1}\\_{k}$. For example: \r\n\r\n$$ r\\_{2}^{−1} \\otimes c = r\\_{2}^{-1} \\otimes \\left(r\\_{1} \\otimes x\\_{1} + r\\_{2} \\otimes x\\_{2} + r\\_{3} \\otimes x\\_{3}\\right) $$\r\n\r\n$$ r\\_{2}^{−1} \\otimes c = x\\_{2} + r^{-1}\\_{2} \\otimes \\left(r\\_{1} \\otimes x\\_{1} + r\\_{3} \\otimes x3\\right) $$\r\n\r\n$$ r\\_{2}^{−1} \\otimes c = x\\_{2} + noise $$\r\n\r\nThe product is exactly $x\\_{2}$ together with a noise term. If the phases of the elements of the key vector are randomly distributed, the noise term has zero mean.\r\n\r\nSource: [Associative LSTMs](https://arxiv.org/pdf/1602.03032.pdf)","description_state":"present","introduced_year":2003,"introduced_by":{"title":null,"paper":null,"first_author":null,"n_authors":0,"url_abs":null,"archive_paper_url":null},"source":{"url":null,"title":null,"url_on_a_paper_host":false},"code_snippet_url":null,"code_snippet_url_on_a_code_host":false,"categories":[{"area":"General","area_id":"general","collection":"Miscellaneous Components","url":"/methods/category/miscellaneous-components","pwc_aliases":[]}],"n_papers_tagged":4,"archive_num_papers":4,"papers_newest_first":[{"paper":null,"title":"Improved Cleanup and Decoding of Fractional Power Encodings","date":"2024-11-30","arxiv_id":"2412.00488","n_code_links":0,"syntology":null},{"paper":null,"title":"Audio Fingerprinting with Holographic Reduced Representations","date":"2024-06-19","arxiv_id":"2406.13139","n_code_links":0,"syntology":null},{"paper":null,"title":"Residual and Attentional Architectures for Vector-Symbols","date":"2022-07-18","arxiv_id":"2207.08953","n_code_links":0,"syntology":null},{"paper":"/paper/associative-long-short-term-memory","title":"Associative Long Short-Term Memory","date":"2016-02-09","arxiv_id":"1602.03032","n_code_links":3,"syntology":null}],"papers_shown":4,"tasks":[{"task":"/task/image-classification","name":"Image Classification","papers":1},{"task":"/task/memorization","name":"Memorization","papers":1},{"task":null,"name":"Position","papers":1},{"task":"/task/retrieval","name":"Retrieval","papers":1},{"task":"/task/image-classification","name":"image-classification","papers":1}],"tasks_shown":5,"n_tasks":5,"usage_by_year":[{"year":"2016","papers":1},{"year":"2022","papers":1},{"year":"2024","papers":2}],"row_source":"methods_table","archive":{"source":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","archive_url":"https://paperswithcode.com/method/holographic-reduced-representation"},"syntology_read_at":"2026-09-24T18:15:14+00:00"}