Gaussian Process Regression
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References: "Gaussian Process", Lectured by Professor Il-Chul Moon "Gaussian Processes", Cornell CS4780 , Lectured by Professor Kilian Weinberger Bayesian Deep Learning by Sungjoon Choi
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The document discusses metric-based few-shot learning approaches. It introduces Matching Networks, which use an attention mechanism to calculate similarity between support and query embeddings. Prototypical Networks determine class membership for a query based on distance to prototype representations of each class. Relation Networks concatenate support and query embeddings and pass them through a relation module to predict relations as classification scores. The approaches aim to learn from few examples by leveraging metric learning in an embedding space.
Sampling method : MCMC
Main obstacles of Bayesian statistics or Bayesian machine learning is computing posterior distribution. In many contexts, computing posterior distribution is intractable. Today, there are two main stream to detour directly computing posterior distribution. One is using sampling method(ex. MCMC) and another is Variational inference. Compared to Variational inference, MCMC takes more time and vulnerable to high-dimensional parameters. However, MCMC has strength in simplicity and guarantees of convergence. I'll briefly introduce several methods people using in application.
Demystifying Neural Style Transfer
Texture synthesis aims to produce new texture samples from an example that are similar but not repetitive. It analyzes the example using a CNN to compute gram matrices representing the texture at different layers, then synthesizes new textures by passing noise through the CNN and minimizing differences from the example's gram matrices. Style transfer extends this to merge the texture of one image onto the content of another by matching gram matrices between layers to transfer style while preserving content. It has been shown that style and content are separable in CNN representations. Style transfer can be viewed as a type of domain adaptation between content and style domains.
Towards Deep Learning Models Resistant to Adversarial Attacks.
This document discusses approaches to training deep neural networks to be robust against adversarial examples. It frames adversarial robustness as a minimax game between the network and an attacker. It presents projected gradient descent (PGD) and the Fast Gradient Sign Method (FGSM) as ways to solve the inner maximization problem during training. Experiments show that adversarially trained models can achieve increased robustness compared to standard networks.
The ways of node embedding
Node embedding techniques learn vector representations of nodes in a graph that can be used for downstream machine learning tasks like classification, clustering, and link prediction. DeepWalk uses random walks to generate sequences of nodes that are treated similarly to sentences, and learns embeddings by predicting nodes using their neighbors, like word2vec. It does not incorporate node features or labels. Node2vec extends DeepWalk by introducing a biased random walk to learn embeddings, addressing some limitations of DeepWalk while maintaining scalability.
Graph Convolutional Network
This document discusses graph convolutional networks (GCNs), which are neural network models for graph-structured data. GCNs aim to learn functions on graphs by preserving the graph's spatial structure and enabling weight sharing. The document outlines the basic components of a GCN, including the adjacency matrix, node features, and application of deep neural network layers. It also notes some challenges with applying convolutions to graphs and discusses approaches like using the graph Fourier transform based on the Laplacian matrix.
Denoising With Frequency Domain
The document discusses different methods for denoising images in the spatial and frequency domains. It introduces spatial domain denoising techniques like mean filtering, median filtering, and adaptive filtering. It then explains how spatial domain images can be transformed into the frequency domain using Fourier and wavelet transforms. This allows denoising based on frequency content, where high frequencies associated with noise can be removed. It concludes by mentioning the CVPR Denoising Workshop as a resource.
Coding Test Review 3
The document contains code snippets and explanations for solving three LeetCode problems: Power of Two, Valid Parentheses, and Find Minimum in Rotated Sorted Array. For Power of Two, it provides an O(log n) solution that uses modulo and division to check if a number is a power of two. For Valid Parentheses, it provides an O(n) solution that uses a string to track opening and closing parentheses. For Find Minimum, it provides both an O(n) solution that finds the minimum by checking if each number is less than the previous, and an O(log n) solution that recursively searches halves of the array to find the minimum.
Time Series Analysis - ARMA
This document provides an overview of time series models and concepts. It discusses stochastic processes, stationarity, the Wold decomposition, impulse response analysis, and ARMA processes. The key points are:
1) Time series models are used to identify shocks and responses over time from stochastic processes.
2) Stationarity assumptions are needed to estimate expectations and variances from time series data using the concept that these values are time-invariant.
3) The Wold decomposition represents a stationary process as the sum of a deterministic component and stochastic prediction errors/shocks.
4) Impulse response analysis examines how past shocks continue to impact the present and future through their effect over time which decays as time
Generative models : VAE and GAN
This document summarizes generative models like VAEs and GANs. It begins with an introduction to information theory, defining key concepts like entropy and maximum likelihood estimation. It then explains generative models as estimating the joint distribution P(X,Y) compared to discriminative models estimating P(Y|X). VAEs are discussed as maximizing the evidence lower bound (ELBO) to estimate the latent variable distribution P(Z|X), allowing generation of new X values. GANs are also covered, defining their minimax game between a generator G and discriminator D, with G learning to generate samples resembling the real data distribution Pemp.
Understanding Blackbox Prediction via Influence Functions
Pang Wei Koh and Percy Liang
"Understanding Black-Box prediction via influence functions" ICML 2017 Best paper
References:
https://youtu.be/0w9fLX_T6tY
https://arxiv.org/abs/1703.04730
Attention Is All You Need
Attention Is All You Need (NIPS 2017)
(Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, Illia Polosukhin)
paper link: https://arxiv.org/pdf/1706.03762.pdf
Reference:
https://youtu.be/mxGCEWOxfe8 (by Minsuk Heo)
https://youtu.be/5vcj8kSwBCY (Stanford CS224N: NLP with Deep Learning | Winter 2019 | Lecture 14 – Transformers and Self-Attention)
Attention
The document discusses different types of attention mechanisms used in neural machine translation and image captioning models. It describes global attention which considers all encoder hidden states when deriving context vectors, and local attention which selectively focuses on a small window of context. Hard attention selects a single location to focus on, while soft attention takes a weighted average over locations. The document also discusses input feeding which makes the model aware of previous alignment choices.
WWW 2020 XAI Tutorial Review
This document is a tutorial on explainable AI from the WWW 2020 conference. It introduces explainable AI and discusses explanations from both a model and regulatory perspective. It then explores different methods for explaining individual predictions, global models, and building interpretable models. The remainder of the tutorial provides case studies on explaining diabetic retinopathy predictions, building an explainable AI engine for talent search, and using model interpretations for sales predictions. References are also included.
Coding test review 2
This document contains summaries of two LeetCode problems - Single Number and Product of Array Except Self.
For Single Number, it provides two O(n) solutions, one using a dictionary to track duplicate numbers and another using math by summing all elements and multiplying by 2, then subtracting the original sum.
For Product of Array Except Self, it again provides two O(n) solutions. The first uses a variable to track the running product and another to count zeros, updating the output array accordingly. The second avoids division by calculating left and right running products in two arrays and multiplying the values together for each output element.
Locality sensitive hashing
This document summarizes the key steps in the locality sensitive hashing (LSH) algorithm for finding similar documents:
1. Documents are converted to sets of shingles (sequences of tokens) to represent them as high-dimensional data points.
2. MinHashing is applied to generate signatures (hashes) for each document such that similar documents are likely to have the same signatures. This compresses the data into a signature matrix.
3. LSH uses the signature matrix to hash similar documents into the same buckets with high probability, finding candidate pairs for further similarity evaluation and filtering out dissimilar pairs from consideration. This improves the computation efficiency over directly comparing all pairs.
Coding Test Review1
This document discusses two algorithms for solving the Two Sum problem from LeetCode: an O(n^2) nested loop solution and an O(n) hash table solution. It also presents a coding interview question to find the maximum prime factor of a given number N and provides a solution using a while loop to iteratively check for divisibility.
SVM (Support Vector Machine & Kernel)
Based on MIT 6.034 Artificial Intelligence, Fall 2010
Instructor: Patrick Winston
video link: https://youtu.be/_PwhiWxHK8o
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Towards Deep Learning Models Resistant to Adversarial Attacks.
Towards Deep Learning Models Resistant to Adversarial Attacks.
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Understanding Blackbox Prediction via Influence Functions
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Highly Motivated architectural engineer with 6 years of experience in interior, exterior, and landscape design, I'm self-motivated person and a competitive professional who is driven by goals with complete dedication and enthusiasm
一比一原版(UQ毕业证书)昆士兰大学毕业证如何办理
原件一模一样【微信：WP101A】【(UQ毕业证书)昆士兰大学毕业证书学位证成绩单】【微信：WP101A】（留信学历认证永久存档查询）采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
【业务选择办理准则】
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二、回国进私企、外企、自己做生意的情况，这些单位是不查询毕业证真伪的，而且国内没有渠道去查询国外文凭的真假，也不需要提供真实教育部认证。鉴于此，办理一份毕业证【微信：WP101A】即可
三、进国企，银行，事业单位，考公务员等等，这些单位是必需要提供真实教育部认证的，办理教育部认证所需资料众多且烦琐，所有材料您都必须提供原件，我们凭借丰富的经验，快捷的绿色通道帮您快速整合材料，让您少走弯路。
留信网认证的作用:
1:该专业认证可证明留学生真实身份【微信：WP101A】
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
→ 【关于价格问题（保证一手价格）
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
选择实体注册公司办理，更放心，更安全！我们的承诺：可来公司面谈，可签订合同，会陪同客户一起到教育部认证窗口递交认证材料，客户在教育部官方认证查询网站查询到认证通过结果后付款，不成功不收费！
办理(UQ毕业证书)昆士兰大学毕业证学位证）学位证【微信：WP101A 】外观非常精致，由特殊纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(UQ毕业证书)昆士兰大学毕业证学位证）学位证【微信：WP101A 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(UQ毕业证书)昆士兰大学毕业证学位证）学位证【微信：WP101A 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(UQ毕业证书)昆士兰大学毕业证学位证）学位证【微信：WP101A 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
FD FAN.pdf forced draft fan for boiler operation and run its very important f...
FD fan or forced draft fan, draws air from the atmosphere and forces it into the furnace through a preheater. These fans are located at the inlet of the boiler to push high pressure fresh air into combustion chamber, where it mixes with the fuel to produce positive pressure. and A forced draft fan (FD fan) is a fan that is used to push air into a boiler or other combustion chamber. It is located at the inlet of the boiler and creates a positive pressure in the combustion chamber, which helps to ensure that the fuel burns properly.
The working principle of a forced draft fan is based on the Bernoulli principle, which states that the pressure of a fluid decreases as its velocity increases. The fan blades rotate and impart momentum to the air, which causes the air to accelerate. This acceleration of the air creates a lower pressure at the outlet of the fan, which draws air in from the inlet.
The amount of air that is pushed into the boiler by the FD fan is determined by the fan’s capacity and the pressure differential between the inlet and outlet of the fan. The fan’s capacity is the amount of air that it can move per unit of time, and the pressure differential is the difference in pressure between the inlet and outlet of the fan.
The FD fan is an essential component of any boiler system. It helps to ensure that the fuel burns properly and that the boiler operates efficiently.
Here are some of the benefits of using a forced draft fan:Improved combustion efficiency: The FD fan helps to ensure that the fuel burns completely, which results in improved combustion efficiency.
Reduced emissions: The FD fan helps to reduce emissions by ensuring that the fuel burns completely.
Increased boiler capacity: The FD fan can increase the capacity of the boiler by providing more air for combustion.
Improved safety: The FD fan helps to improve safety by preventing the buildup of flammable gases in the boiler.
Forced Draft Fan ( Full form of FD Fan) is a type of fan supplying pressurized air to a system. In the case of a Steam Boiler Assembly, this FD fan is of great importance. The Forced Draft Fan (FD Fan) plays a crucial role in supplying the necessary combustion air to the steam boiler assembly, ensuring efficient and optimal combustion processes. Its pressurized airflow promotes the complete and controlled burning of fuel, enhancing the overall performance of the system.What is the FD fan in a boiler?
In a boiler system, the FD fan, or Forced Draft Fan, plays a crucial role in ensuring efficient combustion and proper air circulation within the boiler. Its primary function is to supply the combustion air needed for the combustion process.
The FD fan works by drawing in ambient air and then forcing it into the combustion chamber, creating the necessary air-fuel mixture for the combustion process. This controlled air supply ensures that the fuel burns efficiently, leading to optimal heat transfer and energy production.
In summary, the FD fan i
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UNIT I INCEPTION OF INFORMATION DESIGN 20CDE09-ID
20CDE09- INFORMATION DESIGN
UNIT I INCEPTION OF INFORMATION DESIGN
Introduction and Definition
History of Information Design
Need of Information Design
Types of Information Design
Identifying audience
Defining the audience and their needs
Inclusivity and Visual impairment
Case study.
Analysis and Design of Algorithm Lab Manual (BCSL404)
Analysis and Design of Algorithm Lab Manual (BCSL404)
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Understanding Cybersecurity Breaches: Causes, Consequences, and Prevention
Cybersecurity breaches are a growing threat in today’s interconnected digital landscape, affecting individuals, businesses, and governments alike. These breaches compromise sensitive information and erode trust in online services and systems. Understanding the causes, consequences, and prevention strategies of cybersecurity breaches is crucial to protect against these pervasive risks.
Cybersecurity breaches refer to unauthorized access, manipulation, or destruction of digital information or systems. They can occur through various means such as malware, phishing attacks, insider threats, and vulnerabilities in software or hardware. Once a breach happens, cybercriminals can exploit the compromised data for financial gain, espionage, or sabotage. Causes of breaches include software and hardware vulnerabilities, phishing attacks, insider threats, weak passwords, and a lack of security awareness.
The consequences of cybersecurity breaches are severe. Financial loss is a significant impact, as organizations face theft of funds, legal fees, and repair costs. Breaches also damage reputations, leading to a loss of trust among customers, partners, and stakeholders. Regulatory penalties are another consequence, with hefty fines imposed for non-compliance with data protection regulations. Intellectual property theft undermines innovation and competitiveness, while disruptions of critical services like healthcare and utilities impact public safety and well-being.
Application Infrastructure and cloud computing.pdf
Basic Application Infrastructure and cloud computing.pdf
一比一原版(skku毕业证）韩国成均馆大学毕业证如何办理
原版一模一样【微信：741003700 】【(skku毕业证）韩国成均馆大学毕业证成绩单】【微信：741003700 】学位证，留信学历认证（真实可查，永久存档）原件一模一样纸张工艺/offer、在读证明、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
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【主营项目】
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如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(skku毕业证）韩国成均馆大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(skku毕业证）韩国成均馆大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(skku毕业证）韩国成均馆大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(skku毕业证）韩国成均馆大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
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CONVEGNO DA IRETI 18 giugno 2024 | PASQUALE Donato
"Le potenzialità del Digital Twin per il settore Water"
Coroutines Flow & Channels Workshop Slides
Coroutines Flow & Channels Workshop Slides at Tapsell
YouTube:
https://www.youtube.com/watch?v=SAVRGpgIgXE
Source Code:
https://github.com/MortezaNedaei/Kotlin-Coroutines-Workshop
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FD FAN.pdf forced draft fan for boiler operation and run its very important f...
FD FAN.pdf forced draft fan for boiler operation and run its very important f...
( Call ) Girls Vasant Kunj Just 9873940964 High Class Model Shneha Patil
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Analysis and Design of Algorithm Lab Manual (BCSL404)
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CONVEGNO DA IRETI 18 giugno 2024 | PASQUALE Donato
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Gaussian Process Regression
- 2. Random Process A random process 𝑋𝑡 is completely characterized if the following is known. 𝑃((𝑋𝑡1 , ⋯ ⋯ , 𝑋𝑡 𝑘 ) for any 𝐵, 𝑘, and 𝑡1, ⋯ ⋯ , 𝑡 𝑘 A random process (RP) (or stochastic process) is an infinite indexed collection of random variables {𝑋(𝑡) ∶ 𝑡 ∈ 𝑇 }, defined over a common probability space. (Functions are infinite dimensional vectors) Note that given a random process, only ’finite-dimensional’ probabilities or probability functions can be specified 𝐹𝑜𝑟 𝑡𝑖𝑚𝑒 𝑡 ∈ 𝑇 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔 𝑟𝑎𝑛𝑑𝑜𝑚 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝜔 ∈ Ω 𝑇 × Ω → ℝ
- 5. Gaussian Process A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution
- 7. Gaussian Process * Multivariate and Joint distribution are basically synonyms.
- 9. Gaussian Process Gaussian process and Gaussian process regression are different. Gaussian process regression: A nonparametric Bayesian regression method using the properties of Gaussian processes. Two views to interpret Gaussian process regression • Weight-space view • Function-space view
- 10. MLE vs MAP Linear regression, 𝑓 𝑥 = 𝑤 𝑇 𝑥 𝐺𝑜𝑎𝑙 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑚𝑖𝑛𝑚𝑖𝑧𝑒: 𝑦 − 𝑓(𝑥) 2 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝑤 = (𝑋𝑋 𝑇 )−1 𝑋𝑦
- 11. MLE vs MAP
- 12. MLE vs MAP Another perspective of Bayesian linear regression : Ridge regularization
- 13. MLE vs MAP
- 14. MLE vs MAP
- 15. MLE vs MAP Return to Bayesian solution: Mean value of 𝑥𝑤 𝑀𝐴𝑃
- 16. Gaussian Process regression Gaussian Process regression • Weight Space View • Function Space View
- 17. Weight Space View Gaussian Process regression
- 18. Weight Space View 𝑌 = ∅(𝑥) 𝑇 𝑤, 𝑤~𝑁 0, 𝐴−1 𝐼 𝐼 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑐𝑜𝑙𝑖𝑛𝑒𝑎𝑟𝑡𝑦 𝑖𝑛 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑡𝑒𝑠 𝑜𝑓 𝑤 𝐸 𝑌 = 𝐸 ∅ 𝑥 𝑇 𝑤 = ∅ 𝑥 𝑇 𝐸 𝑤 = 0 Cov 𝑌 = 𝐸 𝑌 − 0 𝑌 − 0 = 𝐸 𝑌𝑌 𝑇 = ∅ 𝑥 𝑇 𝐸 𝑤𝑤 𝑇 ∅ 𝑥 = ∅ 𝑥 𝑇 𝐴−1∅ 𝑥 𝑘 𝑥𝑖, 𝑥𝑗 = 𝑒𝑥𝑝(− 𝑥𝑖 − 𝑥𝑗 2 ) 𝑘 𝑋 𝑇, 𝑋 = ∅ 𝑥 𝑇∅ 𝑥 = 𝑘(𝑥1, 𝑥1) ⋯ 𝑘(𝑥1, 𝑥 𝑛) ⋮ ⋱ ⋮ 𝑘(𝑥 𝑛, 𝑥1) ⋯ 𝑘(𝑥 𝑛, 𝑥 𝑛) 𝑤𝑒 𝑑𝑒𝑓𝑖𝑛𝑒 𝐾 = ∅ 𝑥 𝑇 𝐴−1∅ 𝑥 𝑃 𝑌 = 𝑁(𝑌ㅣ0, 𝐾) 𝑤ℎ𝑎𝑡 𝑑𝑜 𝑤𝑒 𝑔𝑒𝑡 𝑖𝑓 𝑛𝑒𝑤 𝑑𝑎𝑡𝑎 𝑥∗ 𝑎𝑝𝑝𝑒𝑎𝑟? 𝑃 𝑦∗ㅣ𝑥∗, 𝑋, 𝑌 = 𝑁(𝑦∗ㅣ? , ? )
- 19. Weight Space View 𝑤ℎ𝑎𝑡 𝑑𝑜 𝑤𝑒 𝑔𝑒𝑡 𝑖𝑓 𝑛𝑒𝑤 𝑑𝑎𝑡𝑎 𝑥∗ ℎ𝑎𝑠 𝑎𝑝𝑝𝑒𝑎𝑟? 𝑃 𝑦∗ㅣ𝑥∗, 𝑋, 𝑌 = 𝑁(𝑦∗ㅣ? , ? ) 𝑊ℎ𝑎𝑡 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠? 𝑃 𝑦∗ㅣ𝑌
- 20. Weight Space View ∑ 𝑦∗,𝑌 = 𝐸 𝑦∗ 𝑌 𝑇 = ∅(𝑥∗) 𝑇 𝐴−1∅(𝑋) 𝑇 = 𝐶 ∑ 𝑦∗,𝑦∗ = 𝐸 𝑦∗ 𝑦∗ 𝑇 = ∅(𝑥∗) 𝑇 𝐴−1∅(𝑥∗) 𝑇 =k 𝑃 𝑦∗ㅣ𝑌 = 𝑃 ∅(𝑥∗) 𝑇 𝑤ㅣ∅(𝑋) 𝑇 𝑤 = 𝑁(𝑦∗ㅣ𝜇 𝑦∗ + ∑ 𝑦∗,𝑌 ∑ 𝑌,𝑌 −1 𝑌 − 𝜇 𝑌 , ∑ 𝑦∗,𝑦∗ −∑ 𝑦∗,𝑌 ∑ 𝑌,𝑌 −1 ∑ 𝑌,𝑦∗) = 𝑁(𝑦∗ㅣ𝐶𝐾−1 𝑌, 𝑘 − 𝐶𝐾−1 𝐶 𝑇) Var( 𝑌∗)? 𝑌∗ = [𝑦1, ⋯ , 𝑦𝑛, 𝑦 𝑛+1] 𝑇, → Var( 𝑌∗)= 𝑐𝑜𝑣 𝑛 𝐶 𝑇 𝐶 𝑘
- 21. Weight Space View 𝑊ℎ𝑎𝑡 𝑖𝑓 𝑌 = ∅(𝑥) 𝑇 𝑤 + 𝜀, 𝑤~𝑁 0, 𝐴−1 𝐼 , 𝜀~𝑁 0, 𝐵−1 𝐼 Cov 𝑌 = 𝐸 𝑌 − 0 𝑌 − 0 = 𝐸 𝑌𝑌 𝑇 = ∅ 𝑥 𝑇 𝐸 𝑤𝑤 𝑇 ∅ 𝑥 + 𝐸 2𝜀∅ 𝑥 𝑇 𝑤 + 𝜀𝜀 𝑇 = ∅ 𝑥 𝑇 𝐴−1∅ 𝑥 + 𝐵−1 𝐼 = K + 𝐵−1 𝐼
- 22. Function Space View Gaussian Process regression
- 31. Amazing properties of Non-parametric method
- 33. References C. E. Rasmussen and C. K. Williams. Gaussian processes for machine learning, volume 1. MIT press Cambridge, 2006.
- 34. References "Gaussian Process", Lectured by Professor Il-Chul Moon -video link: https://youtu.be/RmN54ykspK4 Ian Goodfellow et al. Deep Learning, (2016) Trevor Hastie et al. The Elements of Statistical Learning (2001) Machine Learning Lecture 26 "Gaussian Processes" -Cornell CS4780 SP17 by Kilian Weinberger -video link: https://www.youtube.com/watch?v=R-NUdqxKjos&t=1000s 9.520/6.860S Statistical Learning Theory by Lorenzo Rosasco http://www.mit.edu/~9.520/fall14/slides/class03/class03_rkhsPart1.pdf -video link: https://www.youtube.com/watch?v=9-oxo_k69qs Bayesian Deep Learning by Sungjoon Choi -video link: https://www.edwith.org/bayesiandeeplearning/joinLectures/14426