Clustering about point (m0, s0) = (0, 2) For N = 2, ⌘W,Z is a valid certificate if |m1 So, one cannot expect to super-resolve variation in the means is too large with Gaussian mixture For x = (m, s) 2 X = R ⇥ R⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2( N = 2 N = 2 N = 2 Clustering about point (m0, s0) = (0, 2). y-axis = me For N = 2, ⌘W,Z is a valid certificate if |m1 m2| 6 |s1 s2| . So, one cannot expect to super-resolve with BLASSO a mixt variation in the means is too large with respect to variations sian mixture = (m, s) 2 X = R ⇥ R⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2(R). N = 2 N = 2 N = 2 N = 2 ring about point (m0, s0) = (0, 2). y-axis = mean, x-axis = standard = 2, ⌘W,Z is a valid certificate if |m1 m2| 6 |s1 s2| . e cannot expect to super-resolve with BLASSO a mixture of 2 Gaussians w on in the means is too large with respect to variations in the standard devia ⇤ + , '(x) = 1 s e (· m)2 2s2 2 H = L 2(R). N = 2 N = 2 N = 2 µ <latexit 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<latexit sha1_base64="kx/AuIG3LNOfw+zA1cXFgxUqzds=">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</latexit> '(x) = 1 e (· µ)2 2 2 <latexit 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<latexit 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<latexit 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Gaussian mixtures: <latexit 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X = R ⇥ R+, <latexit 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<latexit 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<latexit sha1_base64="7UmOgJFSl1i5m9Bfx+DJMCn5ggs=">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</latexit> ⌘W is non-degenerate if m 6 . <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Neuro-imaging Let X = x 2 R2 ; kxk 6 1 . To model MEG/EEG, '(x) = u 7! kx uk 2 2 H = L 2(@X) N = 1 N = 2 N = 2 N = 3 Neuro-imaging: <latexit 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<latexit 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X = x 2 R2 ; ||x|| 6 1 , <latexit 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<latexit 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<latexit 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'(x) = (||x z|| 2)||z||=1 <latexit 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<latexit 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