Scalar products angles and norms x y x y x y n k 1 x k y k dot product x 2 x x

Scalar products angles and norms x y x y x y n k 1 x

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Scalar products, angles and norms x, y = x · y = x * y = n k =1 x k y k (dot product) || x || 2 = x, x = n k =1 x 2 k ( 2 norm) | x, y | || x |||| y || (Cauchy-Schwartz inequality) cos( ( x, y )) = x, y || x |||| y || (angle and cosine) || x + y || 2 = || x || 2 + || y || 2 + 2 x, y (law of cosines) || x || p p = n k =1 | x k | p , p 1 ( p norm) || x + y || p || x || p + || y || p (triangular inequality) Orthogonality, vector space, basis, dimension x y x, y = 0 (Orthogonality) x y ⇔ || x + y || 2 = || x || 2 + || y || 2 (Pythagorean) Let d vectors x i be st x i x j , || x i || = 1 . Define V = Span( { x i } ) = y \ ∃ α C d , y = d i =1 α i x i V is a vector space, { x i } is an orthonormal basis of V and y V, y = d i =1 y, x i x i and d = dim V is called the dimensionality of V . We have dim( V W ) = dim V + dim W - dim( V W ) Column/Range/Image and Kernel/Null spaces Im [ A ] = { y R m \ ∃ x R n such that y = Ax } (image) Ker[ A ] = { x R n \ Ax = 0 } (kernel) Im [ A ] and Ker[ A ] are vector spaces satisfying Im [ A ] = Ker[ A * ] and Ker[ A ] = Im [ A * ] rank A + dim(Ker[ A ]) = n (rank-nullity theorem) where rank A = dim( Im [ A ]) (matrix rank) Note also rank A = rank A * rank A + dim(Ker[ A * ]) = m 19
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How? How? – Piazza If you cannot get access to it contact me asap at [email protected] (title: “ [ECE285-MLIP][Piazza] Access issues ”). 20
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Misc Misc Programming environment: Python/PyTorch/Jupyter We will use UCSD’s DSMLP cluster with GPU/CUDA. Great but busy. We recommend you to install Conda/Python 3/Jupyter on your laptop. Please refer to additional documentations on Piazza. Communication: All your emails must have a title starting with “ [ECE285-MLIP] or it will end up in my spam/trash. Note: “ [ECE 285-MLIP] ”, “ [ece285 MLIP] ”, “ (ECE285MLIP) ” are invalid! But avoid emails, use Piazza to communicate instead. For questions that may interest everyone else, post on Piazza forums. 21
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Some references Reference books C. Bishop Pattern recognition and Machine Learning Springer, 2006 T. Hastie, R. Tibshirani, J. Friedman The Elements of Statistical Learning: Data Mining, Inference, and Prediction Springer, 2009 ~ hastie/ElemStatLearn/ D. Barber Bayesian Reasoning and Machine Learning Cambridge University Press, 2012 I. Goodfellow, Y. Bengio and A. Courville. Deep Learning MIT Press book, 2017 22
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Some references Reference online classes Fei-Fei Li, Justin Johnson and Serena Yeung, 2017 (Stanford) CS231n: Convolutional Neural Networks for Visual Recognition Gir´o et al, 2017 (Catalonia) Deep Learning for Artificial Intelligence Leonardo Araujo dos Santos. Artificial Inteligence 23
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Image sciences
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Imaging sciences – Overview Image sciences Imaging: Modeling the image formation process 24
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Imaging sciences – Overview Image sciences Imaging: Modeling the image formation process Computer graphics: Rendering images/videos from symbolic representation 24
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Imaging sciences – Overview Image sciences Computer vision: Extracting information from images/videos 25
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