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| pg-vector:dev: [1] "nodes": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "node": { | |
| pg-vector:dev: [1] "id": "perceptronArchitecture", | |
| pg-vector:dev: [1] "label": "Perceptron architecture", | |
| pg-vector:dev: [1] "description": "Architecture of a perceptron with one layer of variable connections", | |
| pg-vector:dev: [1] "group": 5, | |
| pg-vector:dev: [1] "citation": "This document discusses the architecture of a perceptron with one layer of variable connections, defining input and information processing neurons, and providing a brief introduction to neural networks." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "node": { | |
| pg-vector:dev: [1] "id": "neurons", | |
| pg-vector:dev: [1] "label": "Types of neurons", | |
| pg-vector:dev: [1] "description": "Input neuron and information processing neuron", | |
| pg-vector:dev: [1] "group": 5, | |
| pg-vector:dev: [1] "citation": "Definition 5.1 (Input neuron) .Anin-input neuron is anidentity neuron . It exactly forwards the information received. Thus, it represents the identity function,input neuron only forwards datawhich should be indicated by the symbol ⧸. Definition 5.2 (Information processing\u00a0neuron) .Information processing neurons somehow process the input infor-mation,i.e. do not represent the identity function. A binary neuron sums up all inputs by using the weighted sum as prop-agation function,which we want to illus-trate by the sign Σ. Then the activation function of the neuron is the binary thresh-old function,which can be illustrated by L|H. This leads us to the complete de-piction of information processing neurons,namely WVUT PQRSΣ L|H. Other neurons that use the weighted sum as propagation function buttheactivation functions hyperbolic tangentorFermi function , or with a sepa-rately defined activation function fact,are similarly represented by WVUT PQRS." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "node": { | |
| pg-vector:dev: [1] "id": "perceptronComponents", | |
| pg-vector:dev: [1] "label": "Components of a perceptron", | |
| pg-vector:dev: [1] "description": "Feedforward network, retina, fixed-weight connections, input layer, information processing layer", | |
| pg-vector:dev: [1] "group": 5, | |
| pg-vector:dev: [1] "citation": "Definition 5.3 (Perceptron) .Theper- ceptron (fig. 5.1 on the facing page) is1a feedforward network containing a retina that is used only for data acquisition and which has fixed-weighted connections with the first neuron layer (input layer). The fixed-weight layer is followed by at least one trainable weight layer. One neuron layer is completely linked with the follow-ing layer. The first layer of the percep-tron consists of the input neurons defined above." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "node": { | |
| pg-vector:dev: [1] "id": "perceptronStructure", | |
| pg-vector:dev: [1] "label": "Structure of a perceptrron", | |
| pg-vector:dev: [1] "description": "One layer of variable connections, input neuron, information processing neuron", | |
| pg-vector:dev: [1] "group": 5, | |
| pg-vector:dev: [1] "citation": "This document discusses the architecture of a perceptron with one layer of variable connections, defining input and information processing neurons, and providing a brief introduction to neural networks." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "node": { | |
| pg-vector:dev: [1] "id": "perceptronArchitectureRelationships", | |
| pg-vector:dev: [1] "label": "Relationships in perceptron architecture", | |
| pg-vector:dev: [1] "description": "Perceptron architecture, components, types of neurons, structure", | |
| pg-vector:dev: [1] "group": 5, | |
| pg-vector:dev: [1] "citation": "This document discusses the architecture of a perceptron with one layer of variable connections, defining input and information processing neurons, and providing a brief introduction to neural networks." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "node": { | |
| pg-vector:dev: [1] "id": "perceptronComponentsRelationships", | |
| pg-vector:dev: [1] "label": "Relationships in perceptron components", | |
| pg-vector:dev: [1] "description": "Fe | |
| pg-vector:dev: [1] ``` | |
| pg-vector:dev: [1] Error with model 'gpt-3.5-turbo-1106': temp=0.1, top_p=0.6, presence_penalty=0.1 at attempt 3 | |
| pg-vector:dev: [1] ####################################################################### | |
| pg-vector:dev: [1] Error: half_json.core import JSONFixer FAILED , got: FixResult(success=False, line='{\n "nodes": [\n {\n "node": {\n "id": "perceptronArchitecture",\n "label": "Perceptron architecture",\n "description": "Architecture of a perceptron with one layer of variable connections",\n "group": 5,\n "citation": "This document discusses the architecture of a perceptron with one layer of variable connections, defining input and information processing neurons, and providing a brief introduction to neural networks."\n }\n },\n {\n "node": {\n "id": "neurons",\n "label": "Types of neurons",\n "description": "Input neuron and information processing neuron",\n "group": 5,\n "citation": "Definition 5.1 (Input neuron) .Anin-input neuron is anidentity neuron . It exactly forwards the information received. Thus, it represents the identity function,input neuron only forwards datawhich should be indicated by the symbol ⧸. Definition 5.2 (Information processing\\u00a0neuron) .Information processing neurons somehow process the input infor-mation,i.e. do not represent the identity function. A binary neuron sums up all inputs by using the weighted sum as prop-agation function,which we want to illus-trate by the sign Σ. Then the activation function of the neuron is the binary thresh-old function,which can be illustrated by L|H. This leads us to the complete de-piction of information processing neurons,namely WVUT PQRSΣ L|H. Other neurons that use the weighted sum as propagation function buttheactivation functions hyperbolic tangentorFermi function , or with a sepa-rately defined activation function fact,are similarly represented by WVUT PQRS."\n }\n },\n {\n "node": {\n "id": "perceptronComponents",\n "label": "Components of a perceptron",\n "description": "Feedforward network, retina, fixed-weight connections, input layer, information processing layer",\n "group": 5,\n "citation": "Definition 5.3 (Perceptron) .Theper- ceptron (fig. 5.1 on the facing page) is1a feedforward network containing a retina that is used only for data acquisition and which has fixed-weighted connections with the first neuron layer (input layer). The fixed-weight layer is followed by at least one trainable weight layer. One neuron layer is completely linked with the follow-ing layer. The first layer of the percep-tron consists of the input neurons defined above."\n }\n },\n {\n "node": {\n "id": "perceptronStructure",\n "label": "Structure of a perceptrron",\n "description": "One layer of variable connections, input neuron, information processing neuron",\n "group": 5,\n "citation": "This document discusses the architecture of a perceptron with one layer of variable connections, defining input and information processing neurons, and providing a brief introduction to neural networks."\n }\n },\n {\n "node": {\n "id": "perceptronArchitectureRelationships",\n "label": "Relationships in perceptron architecture",\n "description": "Perceptron architecture, components, types of neurons, structure",\n "group": 5,\n "citation": "This document discusses the architecture of a perceptron with one layer of variable connections, defining input and information processing neurons, and providing a brief introduction to neural networks."\n }\n },\n {\n "node": {\n "id": "perceptronComponentsRelationships",\n "label": "Relationships in perceptron components",\n "description": "Fe\n```', origin=False) | |
| pg-vector:dev: [1] YAML: ```yaml | |
| pg-vector:dev: [1] nodes: | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitecture | |
| pg-vector:dev: [1] label: Perceptron architecture | |
| pg-vector:dev: [1] description: Architecture of a perceptron with one layer of variable connections | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: neurons | |
| pg-vector:dev: [1] label: Types of neurons | |
| pg-vector:dev: [1] description: Input neuron and information processing neuron | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > Definition 5.1 (Input neuron) .Anin- | |
| pg-vector:dev: [1] > input neuron is anidentity neuron . It | |
| pg-vector:dev: [1] > exactly forwards the information received. | |
| pg-vector:dev: [1] > Thus, it represents the identity function,input neuron | |
| pg-vector:dev: [1] > only forwards | |
| pg-vector:dev: [1] > datawhich should be indicated by the symbol | |
| pg-vector:dev: [1] > ⧸. Therefore the input neuron is repre- | |
| pg-vector:dev: [1] > sented by the symbol GFED @ABC⧸. | |
| pg-vector:dev: [1] > Definition 5.2 (Information processing | |
| pg-vector:dev: [1] > neuron) .Information processing | |
| pg-vector:dev: [1] > neurons somehow process the input infor- | |
| pg-vector:dev: [1] > mation,i.e. do not represent the identity | |
| pg-vector:dev: [1] > function. A binary neuron sums up all | |
| pg-vector:dev: [1] > inputs by using the weighted sum as prop- | |
| pg-vector:dev: [1] > agation function,which we want to illus- | |
| pg-vector:dev: [1] > trate by the sign Σ. Then the activation | |
| pg-vector:dev: [1] > function of the neuron is the binary thresh- | |
| pg-vector:dev: [1] > old function,which can be illustrated by | |
| pg-vector:dev: [1] > L|H. This leads us to the complete de- | |
| pg-vector:dev: [1] > piction of information processing neurons, | |
| pg-vector:dev: [1] > namely WVUT PQRSΣ | |
| pg-vector:dev: [1] > L|H. Other neurons that use the weighted | |
| pg-vector:dev: [1] > sum as propagation function buttheactivation | |
| pg-vector:dev: [1] > functions hyperbolic tangentorFermi | |
| pg-vector:dev: [1] > function , or with a sepa- | |
| pg-vector:dev: [1] > rately defined activation function fact, | |
| pg-vector:dev: [1] > are similarly represented by | |
| pg-vector:dev: [1] > WVUT PQRS | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronComponents | |
| pg-vector:dev: [1] label: Components of a perceptron | |
| pg-vector:dev: [1] description: Feedforward network, retina, fixed-weight connections, input layer, information processing layer | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > Definition 5.3 (Perceptron) .Theper- | |
| pg-vector:dev: [1] > ceptron (fig. 5.1 on the facing page) is1a | |
| pg-vector:dev: [1] > feedforward network containing a retina | |
| pg-vector:dev: [1] > that is used only for data acquisition and | |
| pg-vector:dev: [1] > which has fixed-weighted connections with | |
| pg-vector:dev: [1] > the first neuron layer (input layer). The | |
| pg-vector:dev: [1] > fixed-weight layer is followed by at least | |
| pg-vector:dev: [1] > one trainable weight layer. One neuron | |
| pg-vector:dev: [1] > layer is completely linked with the follow- | |
| pg-vector:dev: [1] > ing layer. The first layer of the percep- | |
| pg-vector:dev: [1] > tron consists of the input neurons defined | |
| pg-vector:dev: [1] > above. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronStructure | |
| pg-vector:dev: [1] label: Structure of a perceptrron | |
| pg-vector:dev: [1] description: One layer of variable connections, input neuron, information processing neuron | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron components | |
| pg-vector:dev: [1] description: Feedforward network, retina, fixed-weight connections, input layer, information processing layer, perceptron architecture | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > Definition 5.3 (Perceptron) .Theper- | |
| pg-vector:dev: [1] > ceptron (fig. 5.1 on the facing page) is1a | |
| pg-vector:dev: [1] > feedforward network containing a retina | |
| pg-vector:dev: [1] > that is used only for data acquisition and | |
| pg-vector:dev: [1] > which has fixed-weighted connections with | |
| pg-vector:dev: [1] > the first neuron layer (input layer). The | |
| pg-vector:dev: [1] > fixed-weight layer is followed by at least | |
| pg-vector:dev: [1] > one trainable weight layer. One neuron | |
| pg-vector:dev: [1] > layer is completely linked with the follow- | |
| pg-vector:dev: [1] > ing layer. The first layer of the percep- | |
| pg-vector:dev: [1] > tron consists of the input neurons defined | |
| pg-vector:dev: [1] > above. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronStructureRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron structure | |
| pg-vector:dev: [1] description: One layer of variable connections, input neuron, information processing neuron, perceptron architecture | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
| pg-vector:dev: [1] description: Perceptron architecture, components, types of neurons, structure, relationships | |
| pg-vector:dev: [1] group: 5 | |
| pg-vector:dev: [1] citation: | | |
| pg-vector:dev: [1] > This document discusses the architecture of a perceptron with one layer of variable connections, | |
| pg-vector:dev: [1] > defining input and information processing neurons, and providing a brief introduction to neural networks. | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] - node: | |
| pg-vector:dev: [1] id: perceptronArchitectureComponentsRelationships | |
| pg-vector:dev: [1] label: Relationships in perceptron architecture and components | |
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| llm-proxy:dev: INFO: 127.0.0.1:39870 - "POST /v1/chat/completions HTTP/1.1" 200 OK | |
| pg-vector:dev: [1] ..!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! SUCCESSFULLY REPAIRED JSON: { | |
| pg-vector:dev: [1] "nodes": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "id": "context", | |
| pg-vector:dev: [1] "label": "Single-layer perceptrons and neural networks", | |
| pg-vector:dev: [1] "description": "Discussing single-layer perceptrons, perceptron learning algorithm, delta rule, and error functions in neural networks", | |
| pg-vector:dev: [1] "group": 3, | |
| pg-vector:dev: [1] "citation": "This document discusses single-layer perceptrons and their use in neural networks, focusing on the perceptron learning algorithm and the delta rule as a gradient-based learning strategy for single-layer perceptrons (SLPs). It also covers the concept of error functions and how they relate to the weights in a neural network.", | |
| pg-vector:dev: [1] "links": [] | |
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| pg-vector:dev: [1] "id": "workHardWhatItEntails", | |
| pg-vector:dev: [1] "label": "Importance of understanding content", | |
| pg-vector:dev: [1] "description": "Discussing the need to understand the context to work effectively with neural networks", | |
| pg-vector:dev: [1] "group": 3, | |
| pg-vector:dev: [1] "citation": "Working hard and what it entails: Putting effort and time into tasks.", | |
| pg-vector:dev: [1] "links": [] | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] ], | |
| pg-vector:dev: [1] "links": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "context", | |
| pg-vector:dev: [1] "target": "workHardWhatItEntails", | |
| pg-vector:dev: [1] "label": "Importance of understanding content", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 7, | |
| pg-vector:dev: [1] "description": "The need to understand the context to work effectively with neural networks", | |
| pg-vector:dev: [1] "citation": "Working hard and what it entails: Putting effort and time into tasks." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] ]} | |
| pg-vector:dev: [1] Success with model 'gpt-3.5-turbo-1106': temp=0.0, top_p=1, presence_penalty=0.0 at attempt 1 | |
| pg-vector:dev: [1] Writing pretty JSON to file: graphs-6d80daa7-b1ca-4074-aa2a-4349da92f47c.json | |
| pg-vector:dev: [1] Contents pretty json: { | |
| pg-vector:dev: [1] "nodes": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "label": "Single-layer perceptrons and neural networks", | |
| pg-vector:dev: [1] "id": "context", | |
| pg-vector:dev: [1] "group": 3, | |
| pg-vector:dev: [1] "citation": "This document discusses single-layer perceptrons and their use in neural networks, focusing on the perceptron learning algorithm and the delta rule as a gradient-based learning strategy for single-layer perceptrons (SLPs). It also covers the concept of error functions and how they relate to the weights in a neural network.", | |
| pg-vector:dev: [1] "links": [] | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "label": "Importance of understanding content", | |
| pg-vector:dev: [1] "id": "workHardWhatItEntails", | |
| pg-vector:dev: [1] "group": 3, | |
| pg-vector:dev: [1] "citation": "Working hard and what it entails: Putting effort and time into tasks.", | |
| pg-vector:dev: [1] "links": [] | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] ], | |
| pg-vector:dev: [1] "links": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "context", | |
| pg-vector:dev: [1] "target": "workHardWhatItEntails", | |
| pg-vector:dev: [1] "label": "Importance of understanding content", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 7, | |
| pg-vector:dev: [1] "citation": "Working hard and what it entails: Putting effort and time into tasks." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] ] | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] Runtime for chunk id 45: 64.54694294929504 seconds | |
| pg-vector:dev: [1] Chunk id 47 metadata: {} others: 238457 9ed51aede4a3fac922afdf460ad7849a317e69551297d58f37bd3080ded9f2bf 4090 672 234368 up these | |
| pg-vector:dev: [1] squares. The summation of the specific er- | |
| pg-vector:dev: [1] rors Errp(W)of all patterns pthen yields | |
| pg-vector:dev: [1] the definition of the error Err and there- | |
| pg-vector:dev: [1] 78 D. Kriesel – A Brief Introduction to Neural Networks (ZETA2-EN) | |
| pg-vector:dev: [1] dkriesel.com 5.1 The singlelayer perceptron | |
| pg-vector:dev: [1] fore the definition of the error function | |
| pg-vector:dev: [1] Err(W): | |
| pg-vector:dev: [1] Err(W) =∑ | |
| pg-vector:dev: [1] p∈PErrp(W) (5.5) | |
| pg-vector:dev: [1] =1 | |
| pg-vector:dev: [1] 2sum over all p | |
| pg-vector:dev: [1] ∑ | |
| pg-vector:dev: [1] p∈P | |
| pg-vector:dev: [1] ∑ | |
| pg-vector:dev: [1] Ω∈O(tp,Ω−yp,Ω)2( | |
| pg-vector:dev: [1] ( | |
| pg-vector:dev: [1] | |
| pg-vector:dev: [1] sum over all Ω. | |
| pg-vector:dev: [1] (5.6) | |
| pg-vector:dev: [1] Theobservantreaderwillcertainlywonder | |
| pg-vector:dev: [1] where the factor1 | |
| pg-vector:dev: [1] 2in equation 5.4 on the | |
| pg-vector:dev: [1] preceding page suddenly came from and | |
| pg-vector:dev: [1] why there is no root in the equation, as | |
| pg-vector:dev: [1] this formula looks very similar to the Eu- | |
| pg-vector:dev: [1] clidean distance. Both facts result from | |
| pg-vector:dev: [1] simple pragmatics: Our intention is to | |
| pg-vector:dev: [1] minimize the error. Because the root func- | |
| pg-vector:dev: [1] tion decreases with its argument, we can | |
| pg-vector:dev: [1] simply omit it for reasons of calculation | |
| pg-vector:dev: [1] and implementation efforts, since we do | |
| pg-vector:dev: [1] not need it for minimization. Similarly, it | |
| pg-vector:dev: [1] does not matter if the term to be mini- | |
| pg-vector:dev: [1] mized is divided by 2: Therefore I am al- | |
| pg-vector:dev: [1] lowed to multiply by1 | |
| pg-vector:dev: [1] 2. This is just done | |
| pg-vector:dev: [1] so that it cancels with a 2in the course of | |
| pg-vector:dev: [1] our calculation. | |
| pg-vector:dev: [1] Now we want to continue deriving the | |
| pg-vector:dev: [1] delta rule for linear activation functions. | |
| pg-vector:dev: [1] We have already discussed that we tweak | |
| pg-vector:dev: [1] the individual weights wi,Ωa bit and see | |
| pg-vector:dev: [1] how the error Err (W)is changing – which | |
| pg-vector:dev: [1] corresponds to the derivative of the er- | |
| pg-vector:dev: [1] ror function Err (W)according to the very | |
| pg-vector:dev: [1] same weight wi,Ω. This derivative cor- | |
| pg-vector:dev: [1] responds to the sum of the derivatives | |
| pg-vector:dev: [1] of all specific errors Err paccording to | |
| pg-vector:dev: [1] this weight (since the total error Err (W)results from the sum of the specific er- | |
| pg-vector:dev: [1] rors): | |
| pg-vector:dev: [1] ∆wi,Ω=−η∂Err(W) | |
| pg-vector:dev: [1] ∂wi,Ω(5.7) | |
| pg-vector:dev: [1] =∑ | |
| pg-vector:dev: [1] p∈P−η∂Errp(W) | |
| pg-vector:dev: [1] ∂wi,Ω.(5.8) | |
| pg-vector:dev: [1] Once again I want to think about the ques- | |
| pg-vector:dev: [1] tion of how a neural network processes | |
| pg-vector:dev: [1] data. Basically, the data is only trans- | |
| pg-vector:dev: [1] ferred through a function, the result of the | |
| pg-vector:dev: [1] function is sent through another one, and | |
| pg-vector:dev: [1] so on. If we ignore the output function, | |
| pg-vector:dev: [1] the path of the neuron outputs oi1andoi2, | |
| pg-vector:dev: [1] which the neurons i1andi2entered into a | |
| pg-vector:dev: [1] neuron Ω, initially is the propagation func- | |
| pg-vector:dev: [1] tion (here weighted sum), from which the | |
| pg-vector:dev: [1] networkinputisgoingtobereceived. This | |
| pg-vector:dev: [1] is then sent through the activation func- | |
| pg-vector:dev: [1] tion of the neuron Ωso that we receive | |
| pg-vector:dev: [1] the output of this neuron which is at the | |
| pg-vector:dev: [1] same time a component of the output vec- | |
| pg-vector:dev: [1] tory: | |
| pg-vector:dev: [1] netΩ→fact | |
| pg-vector:dev: [1] =fact(netΩ) | |
| pg-vector:dev: [1] =oΩ | |
| pg-vector:dev: [1] =yΩ. | |
| pg-vector:dev: [1] As we can see, this output results from | |
| pg-vector:dev: [1] many nested functions: | |
| pg-vector:dev: [1] oΩ=fact(netΩ) (5.9) | |
| pg-vector:dev: [1] =fact(oi1·wi1,Ω+oi2·wi2,Ω).(5.10) | |
| pg-vector:dev: [1] It is clear that we could break down the | |
| pg-vector:dev: [1] output into the single input neurons (this | |
| pg-vector:dev: [1] is unnecessary here, since they do not | |
| pg-vector:dev: [1] D. Kriesel – A Brief Introduction to Neural Networks (ZETA2-EN) 79 | |
| pg-vector:dev: [1] Chapter 5 The perceptron, backpropagation and its variants dkriesel.com | |
| pg-vector:dev: [1] process information in an SLP). Thus, | |
| pg-vector:dev: [1] we want to calculate the derivatives of | |
| pg-vector:dev: [1] equation 5.8 on the preceding page and | |
| pg-vector:dev: [1] due to the nested functions we can apply | |
| pg-vector:dev: [1] thechain rule to factorize the derivative | |
| pg-vector:dev: [1] ∂Errp(W) | |
| pg-vector:dev: [1] ∂wi,Ωin equation 5.8 on the previous | |
| pg-vector:dev: [1] page. | |
| pg-vector:dev: [1] ∂Errp(W) | |
| pg-vector:dev: [1] ∂wi,Ω=∂Errp(W) | |
| pg-vector:dev: [1] ∂op,Ω·∂op,Ω | |
| pg-vector:dev: [1] ∂wi,Ω.(5.11) | |
| pg-vector:dev: [1] Let us take a look at the first multiplica- | |
| pg-vector:dev: [1] tive factor of the above equation 5.11 | |
| pg-vector:dev: [1] which represents the derivative of the spe- | |
| pg-vector:dev: [1] cific error Err p(W)according to the out- | |
| pg-vector:dev: [1] put, i.e. the change of the error Err p | |
| pg-vector:dev: [1] with an output op,Ω: The examination | |
| pg-vector:dev: [1] of Errp(equation 5.4 on page 78) clearly | |
| pg-vector:dev: [1] shows that this change is exactly the dif- | |
| pg-vector:dev: [1] ference between teaching input and out- | |
| pg-vector:dev: [1] put(tp,Ω−op,Ω)(remember: Since Ωis an | |
| pg-vector:dev: [1] output neuron, op,Ω=yp,Ω). The closer | |
| pg-vector:dev: [1] the output is to the teaching input, the | |
| pg-vector:dev: [1] smaller is the specific error. Thus we can | |
| pg-vector:dev: [1] replace one by the other. This difference | |
| pg-vector:dev: [1] is also called δp,Ω(which is the reason for | |
| pg-vector:dev: [1] the name delta rule): | |
| pg-vector:dev: [1] ∂Errp(W) | |
| pg-vector:dev: [1] ∂wi,Ω=−(tp,Ω−op,Ω)·∂op,Ω | |
| pg-vector:dev: [1] ∂wi,Ω | |
| pg-vector:dev: [1] (5.12) | |
| pg-vector:dev: [1] =−δp,Ω·∂op,Ω | |
| pg-vector:dev: [1] ∂wi,Ω(5.13) | |
| pg-vector:dev: [1] The second multiplicative factor of equa- | |
| pg-vector:dev: [1] tion 5.11 and of the following one is the | |
| pg-vector:dev: [1] derivative of the output specific to the pat- | |
| pg-vector:dev: [1] ternpof the neuron Ωaccording to the | |
| pg-vector:dev: [1] weightwi,Ω. So how does op,Ωchange | |
| pg-vector:dev: [1] when the weight from itoΩis changed?Duetotherequirementatthebeginningof | |
| pg-vector:dev: [1] the derivation, we only have a linear acti- | |
| pg-vector:dev: [1] vation function fact, therefore we can just | |
| pg-vector:dev: [1] as well look at the change of the network | |
| pg-vector:dev: [1] input when wi,Ωis Node ID: 634bb180-4c2f-4b38-b856-0a138268dc94 | |
| pg-vector:dev: [1] Text: up these squares. The summation of the specific er- rors | |
| pg-vector:dev: [1] Errp(W)of all patterns pthen yields the definition of the error Err and | |
| pg-vector:dev: [1] there- 78 D. Kriesel – A Brief Introduction to Neural Networks | |
| pg-vector:dev: [1] (ZETA2-EN) dkriesel.com 5.1 The singlelayer perceptron fore the | |
| pg-vector:dev: [1] definition of the error function Err(W): Err(W) =∑ p∈PErrp(W) (5.5) =1 | |
| pg-vector:dev: [1] 2sum over all p ∑ p... | |
| pg-vector:dev: [1] Actual exception: 2 validation errors for KnowledgeGraph | |
| pg-vector:dev: [1] links.6.target | |
| pg-vector:dev: [1] Field required [type=missing, input_value={'source': 'derivationDel...its weight is altered.'}, input_type=dict] | |
| pg-vector:dev: [1] For further information visit https://errors.pydantic.dev/2.5/v/missing | |
| pg-vector:dev: [1] links.7.target | |
| pg-vector:dev: [1] Field required [type=missing, input_value={'source': 'understandDer...elationships involved.'}, input_type=dict] | |
| pg-vector:dev: [1] For further information visit https://errors.pydantic.dev/2.5/v/missing | |
| pg-vector:dev: [1] #################################################### | |
| pg-vector:dev: [1] Got JSON STRING : { | |
| pg-vector:dev: [1] "nodes": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "label": "Derivation of the delta rule for a single-layer perceptron", | |
| pg-vector:dev: [1] "id": "derivationDeltaRule", | |
| pg-vector:dev: [1] "group": null, | |
| pg-vector:dev: [1] "citation": "|The document discusses the derivation of the delta rule for a single-layer perceptron in a neural network. It explains how to calculate the derivatives of the error function and how the output of a neuron changes when its weight is altered.", | |
| pg-vector:dev: [1] "links": [] | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "label": "Understand the derivation of the delta rule", | |
| pg-vector:dev: [1] "id": "understandDerivation", | |
| pg-vector:dev: [1] "group": null, | |
| pg-vector:dev: [1] "citation": "|Understand the derivation of the delta rule for a single-layer perceptron in a neural network to grasp the concepts and relationships involved.", | |
| pg-vector:dev: [1] "links": [] | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] ], | |
| pg-vector:dev: [1] "links": [ | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "derivationDeltaRule", | |
| pg-vector:dev: [1] "target": "understandDerivation", | |
| pg-vector:dev: [1] "label": "Explanation of the derivation", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The context explains the derivation of the delta rule for a single-layer perceptron in a neural network, detailing the various concepts, relationships, and calculations involved in the process", | |
| pg-vector:dev: [1] "citation": "|The document discusses the derivation of the delta rule for a single-layer perceptron in a neural network. It explains how to calculate the derivatives of the error function and how the output of a neuron changes when its weight is altered." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "understandDerivation", | |
| pg-vector:dev: [1] "target": "derivationDeltaRule", | |
| pg-vector:dev: [1] "label": "Understanding the context", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The summary helps to understand the derivation of the delta rule for a single-layer perceptron in a neural network, providing an overview of the concepts and relationships involved in the process", | |
| pg-vector:dev: [1] "citation": "|Understand the derivation of the delta rule for a single-layer perceptron in a neural network to grasp the concepts and relationships involved." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "derivationDeltaRule", | |
| pg-vector:dev: [1] "target": "nodes", | |
| pg-vector:dev: [1] "label": "Nodes in the derivation", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The context contains numerous nodes representing various concepts, relationships, and calculations in the derivation of the delta rule for a single-layer perceptron in a neural network", | |
| pg-vector:dev: [1] "citation": "|The document discusses the derivation of the delta rule for a single-layer perceptron in a neural network, detailing the various nodes and relationships involved in the process." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "understandDerivation", | |
| pg-vector:dev: [1] "target": "nodes", | |
| pg-vector:dev: [1] "label": "Nodes in the understanding", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The summary provides an overview of the numerous nodes involved in understanding the derivation of the delta rule for a single-layer perceptron in a neural network", | |
| pg-vector:dev: [1] "citation": "|Understand the derivation of the delta rule for a single-layer perceptron in a neural network to grasp the concepts and relationships involved, including the numerous nodes that make up the process." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "derivationDeltaRule", | |
| pg-vector:dev: [1] "target": "links", | |
| pg-vector:dev: [1] "label": "Links between nodes", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The context contains numerous links connecting the various nodes in the derivation of the delta rule for a single-layer perceptron in a neural network", | |
| pg-vector:dev: [1] "citation": "|The document discusses the derivation of the delta rule for a single-layer perceptron in a neural network, detailing the various links connecting the nodes and representing the relationships and calculations involved in the process." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "understandDerivation", | |
| pg-vector:dev: [1] "target": "links", | |
| pg-vector:dev: [1] "label": "Links in the understanding", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The summary provides an overview of the numerous links connecting the various nodes involved in understanding the derivation of the delta rule for a single-layer perceptron in a neural network", | |
| pg-vector:dev: [1] "citation": "|Understand the derivation of the delta rule for a single-layer perceptron in a neural network to grasp the concepts and relationships involved, including the numerous links that connect the nodes and represent the relationships and calculations in the process." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "derivationDeltaRule", | |
| pg-vector:dev: [1] "label": "Derivation of the delta rule for a single-layer perceptron", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The context explains the derivation of the delta rule for a single-layer perceptron in a neural network, detailing the various concepts, relationships, and calculations involved in the process", | |
| pg-vector:dev: [1] "citation": "|The document discusses the derivation of the delta rule for a single-layer perceptron in a neural network. It explains how to calculate the derivatives of the error function and how the output of a neuron changes when its weight is altered." | |
| pg-vector:dev: [1] }, | |
| pg-vector:dev: [1] { | |
| pg-vector:dev: [1] "source": "understandDerivation", | |
| pg-vector:dev: [1] "label": "Understand the derivation of the delta rule", | |
| pg-vector:dev: [1] "arrow": "to", | |
| pg-vector:dev: [1] "value": 1, | |
| pg-vector:dev: [1] "description": "The summary helps to understand the derivation of the delta rule for a single-layer perceptron in a neural network, providing an overview of the concepts and relationships involved in the process", | |
| pg-vector:dev: [1] "citation": "|Understand the derivation of the delta rule for a single-layer perceptron in a neural network to grasp the concepts and relationships involved." | |
| pg-vector:dev: [1] } | |
| pg-vector:dev: [1] ] | |
| pg-vector:dev: [1] } |
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