%PDF-1.3 1 0 obj << /Kids [ 3 0 R 4 0 R 5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 10 0 R 11 0 R 12 0 R ] /Type /Pages /Count 10 >> endobj 2 0 obj << /Title (Learning Invariant Riemannian Geometric Representations Using Deep Nets) /Producer (PyPDF2) /Author (Suhas Lohit\054 Pavan Turaga) /Subject (2017 IEEE International Conference on Computer Vision) >> endobj 3 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 14 0 R /Resources << /XObject << /x8 15 0 R /x6 18 0 R /x12 21 0 R /x10 24 0 R >> /ExtGState << /s9 27 0 R /s11 30 0 R /R27 33 0 R /a0 << /CA 1 /ca 1 >> /s5 34 0 R /s7 37 0 R >> /Font << /R46 40 0 R /R44 44 0 R /R42 47 0 R /R40 52 0 R /F2 56 0 R /R28 57 0 R /F1 61 0 R /R38 62 0 R /R36 66 0 R /R34 70 0 R /R32 74 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 81 0 R 82 0 R 83 0 R 84 0 R 85 0 R 86 0 R 87 0 R 88 0 R 89 0 R 90 0 R 91 0 R 92 0 R 93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R ] >> endobj 4 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 99 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /R46 40 0 R /R44 44 0 R /R42 47 0 R /F2 100 0 R /R28 57 0 R /F1 101 0 R /R38 62 0 R /R36 66 0 R /R34 70 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 102 0 R 103 0 R 104 0 R 105 0 R 106 0 R 107 0 R 108 0 R 109 0 R 110 0 R 111 0 R 112 0 R 113 0 R 114 0 R 115 0 R 116 0 R 117 0 R ] >> endobj 5 0 obj << /Parent 1 0 R /Rotate 0 /Contents 118 0 R /Resources << /ColorSpace << /R89 119 0 R >> /XObject << /R112 120 0 R /R109 123 0 R /R110 124 0 R /R111 125 0 R /R104 126 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] /Pattern << /R92 127 0 R >> /ExtGState << /R88 130 0 R /R83 131 0 R /R82 132 0 R /R103 121 0 R /R93 133 0 R /R27 33 0 R >> /Font << /R46 40 0 R /R44 44 0 R /R42 47 0 R /R127 134 0 R /R99 138 0 R /R125 145 0 R /F1 148 0 R /R123 149 0 R /R121 152 0 R /R106 156 0 R /R95 162 0 R /R85 169 0 R /F2 176 0 R /R119 177 0 R /R113 180 0 R /R117 184 0 R /R115 187 0 R /R28 57 0 R /R38 62 0 R /R36 66 0 R /R34 70 0 R /R32 74 0 R /R30 77 0 R >> /Shading << /R91 128 0 R >> >> /Group 191 0 R /MediaBox [ 0 0 612 792 ] /Annots [ 192 0 R 193 0 R 194 0 R 195 0 R 196 0 R 197 0 R 198 0 R ] /Type /Page >> endobj 6 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 199 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /R144 200 0 R /R146 205 0 R /R150 209 0 R /R46 40 0 R /R44 44 0 R /R42 47 0 R /R148 213 0 R /F2 216 0 R /R28 57 0 R /F1 217 0 R /R38 62 0 R /R36 66 0 R /R34 70 0 R /R40 52 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 218 0 R 219 0 R 220 0 R 221 0 R 222 0 R 223 0 R ] >> endobj 7 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 224 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /R144 200 0 R /R146 205 0 R /R167 225 0 R /R150 209 0 R /R165 228 0 R /R46 40 0 R /R44 44 0 R /R42 47 0 R /R40 52 0 R /F2 231 0 R /R28 57 0 R /F1 232 0 R /R38 62 0 R /R36 66 0 R /R32 74 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 233 0 R 234 0 R 235 0 R 236 0 R 237 0 R 238 0 R 239 0 R 240 0 R ] >> endobj 8 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 241 0 R /Resources << /XObject << /R192 242 0 R /R193 243 0 R /R190 244 0 R /R191 245 0 R /R185 246 0 R /R184 247 0 R /R180 248 0 R /R189 249 0 R /R188 250 0 R /R181 251 0 R /R183 252 0 R /R182 253 0 R /R178 254 0 R /R179 255 0 R >> /ExtGState << /R27 33 0 R >> /Font << /R119 177 0 R /R186 256 0 R /R40 52 0 R /R46 40 0 R /R42 47 0 R /R117 184 0 R /F2 259 0 R /R28 57 0 R /F1 260 0 R /R38 62 0 R /R36 66 0 R /R115 187 0 R /R30 77 0 R >> /ProcSet [ /Text /ImageC /ImageB /PDF /ImageI ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 261 0 R 262 0 R 263 0 R 264 0 R ] >> endobj 9 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 265 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /R144 200 0 R /R201 266 0 R /R167 225 0 R /R150 209 0 R /R165 228 0 R /R46 40 0 R /F1 269 0 R /R44 44 0 R /R42 47 0 R /R148 213 0 R /F2 270 0 R /R28 57 0 R /R146 205 0 R /R38 62 0 R /R36 66 0 R /R40 52 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 271 0 R ] >> endobj 10 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 272 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /R144 200 0 R /R201 266 0 R /R214 273 0 R /R165 228 0 R /R46 40 0 R /F1 276 0 R /R42 47 0 R /R117 184 0 R /F2 277 0 R /R28 57 0 R /R146 205 0 R /R38 62 0 R /R36 66 0 R /R148 213 0 R /R115 187 0 R /R32 74 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ 278 0 R 279 0 R 280 0 R 281 0 R 282 0 R 283 0 R ] >> endobj 11 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 284 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /R113 180 0 R /F2 285 0 R /R28 57 0 R /F1 286 0 R /R34 70 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ ] >> endobj 12 0 obj << /Parent 1 0 R /Rotate 0 /Type /Page /Contents 287 0 R /Resources << /ExtGState << /R27 33 0 R >> /Font << /F2 288 0 R /R34 70 0 R /F1 289 0 R /R30 77 0 R >> /ProcSet [ /ImageC /Text /PDF /ImageI /ImageB ] >> /MediaBox [ 0 0 612 792 ] /Annots [ ] >> endobj 13 0 obj << /Type /Catalog /Pages 1 0 R >> endobj 14 0 obj << /Length 22017 >> stream q q q 0.10000 0 0 0.10000 0 0 cm /R27 gs 0 g q 10 0 0 10 0 0 cm BT /R28 14.34620 Tf 1 0 0 1 62.50900 675.06700 Tm [ (Lear) 14.98930 (ning) -249.99400 (In) 40.00800 (v) 9.99625 (ariant) -250.00300 (Riemannian) -250.00300 (Geometric) -249.99000 (Repr) 18.00140 (esentations) -249.99800 (Using) -249.99100 (Deep) -250.00800 (Nets) ] TJ /R30 11.95520 Tf 126.51500 -37.85820 Td [ (Suhas) -249.98700 (Lohit) ] TJ -30.46680 -13.94690 Td [ (Arizona) -249.99800 (State) -250.00200 (Uni) 24.99570 (v) 14.98510 (ersity) ] TJ /R32 8.96640 Tf 21.53590 -13.94800 Td (slohit\100asu\056edu) Tj /R30 11.95520 Tf 165.00500 27.89490 Td [ (P) 15.01580 (a) 19.98840 (v) 24.98340 (an) -250.00800 (T) 44.98810 (urag) 4.98688 (a) ] TJ -26.82110 -13.94690 Td [ (Arizona) -250 (State) -250.00200 (Uni) 24.99570 (v) 14.98510 (ersity) ] TJ /R32 8.96640 Tf 18.84610 -13.94800 Td (pturaga\100asu\056edu) Tj /R28 11.95520 Tf -191.12800 -41.04610 Td (Abstract) Tj /R34 9.96260 Tf -83.92770 -24.32810 Td [ (Non\055Euclidean) -396.98200 (constr) 15.00240 (aints) -397.00100 (ar) 36.98520 (e) -396.99700 (inher) 36.99380 (ent) -396.98500 (in) -397.02000 (many) -396.98500 (kinds) ] TJ -11.95510 -11.95510 Td [ (of) -282.01900 (data) -282.99000 (in) -282.01900 (computer) -282.99000 (vision) -282.00600 (and) -282.99700 (mac) 14.98160 (hine) -282.01900 (learning) 9.99588 (\054) -291.00800 (typically) ] TJ 11.95470 TL T* [ (as) -294.01800 (a) -293.98500 (r) 37.01960 (esult) -294.00900 (of) -292.99800 <73706563690263> -294.02000 (in) 40.00560 (variance) -294.01500 (r) 37.01960 (equir) 36.99380 (ements) -294.01700 (that) -294.00500 (need) -293.98800 (to) ] TJ 11.95630 TL T* [ (be) -283.01100 (r) 37.01960 (es) 0.98207 (pected) -283.00700 (during) -283.01800 (high\055l) 1.00596 (e) 13.99280 (vel) -281.99300 (infer) 36.99510 (ence) 15.00980 (\056) -408.00800 (Often\054) -290.01000 (these) -283.00200 (g) 10.00320 (eo\055) ] TJ 11.95470 TL T* [ (metric) -255.98200 (constr) 15.00240 (aints) -256.98500 (can) -257.01000 (be) -255.99500 (e) 19.99180 (xpr) 36.98160 (essed) -257.00700 (in) -255.98400 (the) -257.00900 (langua) 9.99098 (g) 10.00320 (e) -256.00100 (of) -257.00400 (Rie\055) ] TJ T* [ (mannian) -400.99600 (g) 10.00320 (eometry) 55.01900 (\054) -438.98300 (wher) 36.99380 (e) -400.99700 (con) 40 (ventional) -401 (vector) -400.98700 (space) -400.98500 (ma\055) ] TJ T* [ (c) 15.01280 (hine) -358.00600 (learning) -359.00300 (does) -358.00600 (not) -358.00200 (apply) -358.98100 (dir) 36.99260 (ectly) 55.01290 (\056) -634.99000 (The) -358.00400 (centr) 14.98770 (al) -358.98700 (ques\055) ] TJ T* [ (tion) -252.99200 (this) -254.01000 (paper) -253 (deals) -254.01800 (with) -252.98400 (is\072) ] TJ /R30 9.96260 Tf 117.08300 0 Td [ (Ho) 24.98360 (w) -253.99400 (does) -253.00400 (one) -253.01200 (train) -253.98500 (deep) -253.01400 (neu\055) ] TJ -117.08300 -11.95510 Td [ (ral) -301.98900 (nets) -302.01400 (whose) -302.01300 <026e616c> -300.98700 (outputs) -301.98800 (are) -302.00100 (elements) -302.00400 (on) -302.01800 (a) -301.99400 (Riemannian) ] TJ 11.95590 TL (manifold\077) ' /R34 9.96260 Tf 48.09880 0 Td [ (T) 92 (o) -404.00600 (answer) -405 (this\054) -442.01600 (we) -404.98300 (pr) 44.98390 (opose) -403.99400 (a) -404.00600 (g) 10.00320 (ener) 15.01960 (al) -405.01800 (fr) 14.99010 (ame\055) ] TJ -48.09880 -11.95510 Td [ (work) -302.00800 (for) -302.01000 (manifold\055awar) 37.00610 (e) -301.99400 (tr) 14.99140 (aining) -301.99400 (of) -302.01600 (deep) -301.98600 (neur) 14.99010 (al) -302.01600 (networks) ] TJ 11.95470 TL T* [ (\226) -418.00400 (we) -419.01900 (utilize) -418.01200 (tang) 9.98118 (ent) -418.00100 (spaces) -417.99500 (and) -419.01300 (e) 19.99180 (xponential) -418.01100 (maps) -417.98900 (in) -419.01600 (or) 20.00890 (\055) ] TJ T* [ (der) -384.00800 (to) -384.98200 (con) 39.99880 (vert) -384.00900 (the) -384.98700 (pr) 44.98390 (oposed) -383.99200 (pr) 44.98390 (oblem) -385.01800 (into) -383.99000 (a) -384.99000 (form) -383.99200 (that) -385.00900 (al\055) ] TJ T* [ (lows) -274.98200 (us) -275.00200 (to) -275.98100 (bri) 0.99861 (ng) -275.98300 (curr) 36.98650 (ent) -275.00600 (advances) -275 (in) -275 (deep) -275.01000 (learning) -275.99600 (to) -275 (bear) ] TJ T* [ (upon) -357.01800 (this) -355.99300 (pr) 44.98390 (oblem\056) -629.99400 (W) 91.98590 (e) -355.98500 (describe) -356.98300 (two) -355.99300 <73706563690263> -356.98900 (applications) ] TJ 11.95590 TL T* [ (to) -233.98800 (demonstr) 15.01100 (ate) -233.01300 (this) -234.01400 (appr) 44.99370 (oac) 14.98400 (h\072) -301.98800 (pr) 36.98520 (ediction) -232.99300 (of) -233.98800 (pr) 44.98510 (obability) -233.99500 (dis\055) ] TJ 11.95510 TL T* [ (trib) 19.98320 (utions) -327.98800 (for) -326.98600 (multi\055class) -328.01600 (ima) 10.01300 (g) 10.00320 (e) -327.00900 <636c6173736902636174696f6e2c> -347.02100 (and) -328.00900 (pr) 36.98650 (edic\055) ] TJ T* [ (tion) -248.99300 (of) -249.00500 (illumination\055in) 39.99460 (variant) -248.98700 (subspaces) -248.98800 (fr) 44.98510 (om) -249.01000 (a) -249.01200 (single) -248.99000 (face\055) ] TJ T* [ (ima) 10.01360 (g) 10.00320 (e) -265.01900 (via) -265.98700 (r) 37.01960 (e) 39.98900 (gr) 36.98520 (ession) -264.99000 (on) -265.00500 (the) -265.00700 (Gr) 14.98650 (assmannian\056) -356.00400 (These) -266.01700 (applica\055) ] TJ T* [ (tions) -339.01100 (show) -337.99300 (the) -338.99500 (g) 10.00320 (ener) 15.01960 (ality) -338 (of) -338.99000 (the) -338.01500 (pr) 44.98390 (oposed) -338.98000 (fr) 14.99010 (ame) 14.98030 (work\054) -361.01600 (and) ] TJ 11.95590 TL T* [ (r) 37.01960 (esult) -320.98400 (in) -320.01300 (impr) 44.99370 (o) 10.00320 (ved) -321.00900 (performance) -319.99600 (o) 10.00320 (ver) -321.00800 (baselines) -319.99300 (that) -320.98100 (ignor) 37.00360 (e) ] TJ 11.95510 TL T* [ (the) -235.99200 (g) 10.00320 (eometry) -237.01100 (of) -235.98800 (the) -237.01200 (output) -236.00500 (space) 14.98520 (\056) -305.98600 (In) -235.99900 (addition) -237.01200 (to) -235.98800 (solving) -236.98700 (this) ] TJ T* [ <73706563690263> -318.01500 (pr) 44.98390 (oblem\054) -334.00400 (we) -316.99700 (belie) 14.98890 (ve) -317.99500 (this) -318 (paper) -316.98900 (opens) -318.00800 (ne) 15.01710 (w) -316.99400 (lines) -318.00300 (of) ] TJ T* [ (enquiry) -284.01100 (center) 37.01830 (ed) -283.01000 (on) -283.98200 (the) -283.98400 (implications) -282.99100 (of) -284.01900 (Riemannian) -283.98900 (g) 10.00320 (eom\055) ] TJ T* [ (etry) -249.99300 (on) -249.98800 (deep) -249.99600 (ar) 36.98520 (c) 15.01220 (hitectur) 37.00360 (es\056) ] TJ /R28 11.95520 Tf 37.11800 TL T* [ (1\056) -249.99000 (Intr) 18.01460 (oduction) ] TJ /R30 9.96260 Tf 11.95510 -19.34570 Td [ (Man) 14.98950 (y) -315.00100 (applications) -315.00400 (in) -316.01300 (computer) -314.98400 (vision) -315.01100 (emplo) 9.98363 (y) -315.00100 (data) -315.98400 (that) ] TJ -11.95510 -11.95630 Td [ (are) -222.99400 (naturally) -222.01700 (represented) -223.01700 (on) -221.99200 (manifolds) -222.98100 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 207.83000 188.59600 Tm (32) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 217.79300 188.59600 Tm [ (\135\056) -301.00900 (Shapes) -222.01900 (that) -223.00700 (are) ] TJ -167.68000 -11.95510 Td [ (in) 40.00560 (v) 24.98110 (ariant) -281.98500 (to) -281 (af) 25.00810 <026e65> -282.01700 (transforms) -280.98500 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 172.21400 176.64100 Tm (5) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 177.19500 176.64100 Tm [ (\135) -281.99600 (and) -281.98600 (linear) -281.00500 (dynamical) -282.00200 (sys\055) ] TJ -127.08300 -11.95470 Td [ (tems) -193.98600 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 74.18160 164.68600 Tm (42) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 84.14410 164.68600 Tm [ (\135) -194.01000 (can) -194.01100 (be) -194.00600 (represented) -194.00200 (as) -194.00600 (points) -194.99100 (on) -193.99700 (the) -194 (Grassmannian\056) ] TJ -34.03200 -11.95510 Td [ (In) -283.99000 (dif) 24.98540 (fusion) -284.00600 (tensor) -284.01500 (imaging\054) -291.99000 (each) -284 (\224pix) 14.99010 (el\224) -283.99400 (of) -283.98900 (the) -283.98400 (\224image\224) -283.98400 (is) ] TJ 11.95510 TL T* [ (a) -365.00200 (symmetric) -366.01900 (positi) 25.01420 (v) 14.98280 (e) -365.00300 <6465026e697465> -365.00700 (\050SPD\051) -365.99300 (matrix) -365.01300 (and) -364.99300 (the) -366.01000 (space) ] TJ T* [ (of) -386.01200 (SPD) -387.00200 (matrices) -385.99800 (forms) -385.99700 (a) -387 (manifold) -386.01900 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 200.48800 128.82100 Tm (36) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 210.45000 128.82100 Tm [ (\135\056) -719.01700 (Lie) -386.00700 (groups) -387.01100 (lik) 10.01790 (e) ] TJ /R36 9.96260 Tf -160.33800 -11.95590 Td [ (S) -58.18210 (O) -0.79950 ] TJ /R38 9.96260 Tf 14.55900 0 Td [ (\050) -0.89997 (3\051) -0.89997 ] TJ /R30 9.96260 Tf 16.28590 0 Td (and) Tj /R36 9.96260 Tf 17.94220 0 Td [ (S) -58.18270 (E) -0.20095 ] TJ /R38 9.96260 Tf 14.61090 0 Td [ (\050) -0.89936 (3\051) -0.90058 ] TJ /R30 9.96260 Tf 16.28590 0 Td [ (are) -357.01200 (used) -356.98700 (to) -356.98700 (represent) -356.99400 (human) -355.99700 (sk) 10.01060 (eletons) ] TJ -79.68400 -11.95510 Td (\133) Tj ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 53.42970 104.91000 Tm (43) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 63.39220 104.91000 Tm (\054) Tj ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 68.31370 104.91000 Tm (44) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 78.27620 104.91000 Tm [ (\135\056) -307.98800 (Predicting) -243.99900 (probabilit) 0.98268 (y) -243.99400 (density) -244.00600 (functions) -244.00600 (is) -243.98600 (another) ] TJ -28.16410 -11.95510 Td [ (area) -294.98700 (of) -295.98800 (interest\054) -306.00400 (applicable) -296.00900 (to) -294.99700 (multi\055class) -295.00200 <636c6173736902636174696f6e> -295.98000 (and) ] TJ 11.95510 TL T* [ (bag\055of\055w) 10.00200 (ords) -249.98500 (models) -250.00600 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 140.76200 81 Tm (30) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 150.72400 81 Tm [ (\135\054) -249.99900 (and) -249.99100 (salienc) 14.99010 (y) -249.99300 (prediction) -250.00700 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 257.09500 81 Tm (25) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 267.05700 81 Tm (\135\056) Tj 53.75980 487.26800 Td [ (Se) 24.98600 (v) 14.98280 (eral) -448.00900 (years) -448.00400 (of) -448.98100 (research) -448.01800 (has) -447.99900 (presented) -447.99400 (us) -449.01300 (wi) 0.99003 (th) -449.01300 (v) 24.98110 (ari\055) ] TJ -11.95510 -11.95630 Td [ (ous) -387.01600 (tools) -387.98400 (for) -386.99700 (statistics\054) -421.99800 (and) -386.98900 (thereby) -388.00400 (machine) -387.01900 (learning) -387.99400 (ap\055) ] TJ 11.95510 TL T* [ (proaches) -326.01900 (to) -324.99200 (be) -325.98500 (deplo) 10.01550 (yed) -326.01900 (when) -324.99500 (the) -326.01400 (objects) -324.98500 (of) -325.98500 (interest) -326.00400 (ha) 19.99670 (v) 14.98280 (e) ] TJ T* [ (manifold\055v) 25.01300 (alued) -273.98600 (domains) -273.01800 (\050c\056f\056) -381 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 436.83200 532.40200 Tm (40) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 446.79400 532.40200 Tm [ (\135\051\056) -381 (In) -273.99300 (deep) -273.99300 (l) 0.98513 (earning\054) -280.01700 (it) -274.01800 (is) ] TJ -137.93200 -11.95510 Td [ (usually) -335.99100 (the) -334.99600 (case) -335.99100 (that) -336.00800 (data) -336.01700 (samples) -335.01300 (are) -335.99300 (vie) 24.98600 (wed) -336.01700 (as) -335.98300 (elements) ] TJ 11.95510 TL T* [ (of) -438.00300 (v) 14.98280 (ector) -438.98600 (spaces\056) -874.00400 (An) 14.98520 (y) -438.98100 (additional) -438.00500 (structure) -437.99100 (that) -437.99100 (the) -439.01500 (data) ] TJ 11.95590 TL T* [ (may) -239.98700 (possess) -240.00900 (is) -239.98700 (left) -239.01200 (to) -239.98700 (be) -240 (learned) -240 (through) -240.00900 (the) -239.98900 (training) -240.00900 (e) 15.01220 (xam\055) ] TJ 11.95510 TL T* [ (ples\056) -399.01400 (Ho) 24.98360 (we) 25.01300 (v) 14.98280 (er) 39.98600 (\054) -287.00800 (recently) 65.01610 (\054) -287.00800 (there) -279.99200 (has) -279.98800 (been) -279.01200 (interest) -280.01200 (in) -280.02200 (emplo) 9.98363 (y\055) ] TJ T* [ (ing) -387.99700 (deep) -388.99100 (l) 0.98758 (earning) -388.98400 (techniques) -387.98400 (for) -388.01400 (non\055Euclidean) -387.97900 (inputs) -388.99900 (as) ] TJ T* [ (well\072) -300.98900 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 335.10400 460.67100 Tm (7) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 340.08500 460.67100 Tm [ (\135) -232.98300 (including) -233.01700 (graph\055structured) -232.01500 (data) -233.01500 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 476.07500 460.67100 Tm (8) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 481.05600 460.67100 Tm (\054) Tj ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 485.86800 460.67100 Tm (20) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 495.83000 460.67100 Tm (\054) Tj ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 500.63200 460.67100 Tm (34) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 510.59500 460.67100 Tm [ (\135) -232.98600 (and) -233.01500 (3D) ] TJ -201.73300 -11.95470 Td [ (shapes) -332.98900 (vie) 24.98360 (wed) -333 (as) -331.98100 (Riemannian) -332.99800 (manifolds) -333.00300 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 480.13900 448.71600 Tm (33) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 490.10200 448.71600 Tm [ (\135\056) -557.98500 (Also\054) -353.99500 (deep) ] TJ -181.23900 -11.95510 Td [ (netw) 10.00810 (orks) -191.99500 (that) -191.01200 (preserv) 15.01710 (e) -192.01200 (the) -191.99700 (input) -190.99800 (geometry) -191.98300 (at) -190.98300 (each) -192.01700 (layer) -192.00700 (ha) 19.99670 (v) 14.98280 (e) ] TJ 11.95630 TL T* [ (been) -416.00700 (studied) -416.01600 (for) -416.99200 (inference) -416.01400 (problems\054) -458.00700 (e\056g\056\054) -457.01200 (for) -416.98900 (symmetric) ] TJ 11.95470 TL T* [ (positi) 25.01300 (v) 14.98280 (e) -331.98900 <6465026e697465> -331.01300 (matrices) -332.00800 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 416.88700 412.85000 Tm (22) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 426.84900 412.85000 Tm [ (\135\054) -351.98000 (Li) 0.99003 (e) -331.98900 (groups) -331.99900 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 486.48500 412.85000 Tm (23) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 496.44800 412.85000 Tm [ (\135) -331.00800 (and) -332.01800 (points) ] TJ -187.58600 -11.95510 Td [ (on) -308.01700 (the) -309 (Stiefel) -308.00300 (manifold) -309.01200 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 408.59800 400.89500 Tm (24) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 418.56100 400.89500 Tm [ (\135\056) -485.01700 (Another) -308.98800 (recent) -307.99300 (w) 10.00320 (ork) -309.00200 (consid\055) ] TJ -109.69800 -11.95510 Td [ (ers) -232.00300 (weight) -232.01800 (matrices) -233.00300 (which) -231.98600 (are) -232.01000 (constrained) -232.01500 (to) -231.99100 (be) -233.02000 (orthogonal\054) ] TJ 11.95510 TL T* [ (i\056e\056\054) -340.99200 (points) -321.99000 (of) -323.00200 (the) -322.99800 (Stiefel) -322 (manifold\054) -341.00700 (and) -322.02000 (propose) -322.98500 (a) -323.01000 (general\055) ] TJ T* [ (ized) -305.00500 (v) 14.98280 (ersion) -304.99300 (of) -305.00500 (backpropag) 4.99426 (ation) -304.98800 (\133) ] TJ ET Q 0 1 0 rg q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 444.76200 365.03000 Tm (19) Tj ET Q 0 g q 10 0 0 10 0 0 cm BT /R30 9.96260 Tf 1 0 0 1 454.72500 365.03000 Tm [ (\135\056) -475.01900 (These) -305.00300 (w) 10.00320 (orks) -304.99300 (do) -304.99800 (not) ] TJ -145.86300 -11.95630 Td [ (consider) -249.98000 (output) -250.00200 (v) 24.98110 (ariables) -249.98800 (with) -250.01200 (geometric) -250.01700 (constraints\056) ] TJ 11.95510 -12.18010 Td [ (In) -327.00200 (contrast) -326.00400 (to) -326.99200 (the) -326.01700 (abo) 14.98280 (v) 14.98280 (e\054) -345.98100 (instead) -326.98500 (of) -325.98500 (enforcing) -326.99400 (geometry) ] TJ -11.95510 -11.95470 Td [ (at) -311.98500 (the) -312.01900 (inputs\054) -327.00700 (our) -312.01900 (goal) -312.01400 (is) -312.01400 (to) -312.01700 (design) -312.00200 (a) -311.99200 (general) -311.98700 (frame) 25.01050 (w) 10.00320 (ork) -312.02100 (to) ] TJ 11.95630 TL T* [ (e) 15.01220 (xtend) -266.99700 (neural) -267.99000 (netw) 10.00810 (ork) -267.00700 (architectures) -267.00400 (where) -267.01900 (output) -268 (v) 24.98110 (ariables) ] TJ 11.95470 TL T* [ (\050or) -285.99400 (deeper) -287.01800 (feature) -285.99900 (maps\051) -286.99600 (lie) -285.98200 (on) -285.98200 (manifolds) -287.01100 (of) -285.99100 (kno) 24.99090 (wn) -287.00100 (geom\055) ] TJ T* [ (etry) 65.01370 (\054) -364.01500 (typically) -341.00700 (due) -340.99700 (to) -340.99000 (certain) -340.99700 (in) 40.00310 (v) 24.98110 (ariance) -341.01200 (requirements\056) -583.01100 (W) 79.98660 (e) ] TJ T* [ (do) -231.01000 (not) -231.00300 (assume) -231 (the) -229.99300 (inputs) -230.98600 (themselv) 15.01220 (es) -231.02000 (ha) 19.99670 (v) 14.98280 (e) -230.98600 (kno) 24.99090 (wn) -231.01000 (geometric) ] TJ T* [ (structure) -344.98700 (and) -344.01600 (emplo) 9.98363 (y) -344.99600 (standard) -344.00600 (back\055propag) 4.98936 (ation) -344.98200 (for) -345.00100 (train\055) ] TJ T* [ (ing\056) -310.00700 (Equi) 24.99830 (v) 24.98110 (alently) 64.99160 (\054) -249.01500 (one) -249.01200 (may) -249.00300 (consider) -249.00300 (this) -248.99300 (approach) -249.99300 (as) -249.01700 (trying) ] TJ 11.95630 TL T* [ (to) -223.99000 (estimate) -224.01400 (a) -224.00700 (mapping) -224.00700 (from) -223.01200 (an) -224.00200 (input) ] TJ /R40 9.96260 Tf 145.11700 0 Td [ (x) -0.89691 ] TJ /R42 9.96260 Tf 8.81406 0 Td [ (2) -0.69596 ] TJ /R44 9.96260 Tf 9.40898 0 Td [ (R) -0.19605 ] TJ /R46 6.97380 Tf 7.19492 3.61602 Td [ (N) -0.90321 ] TJ /R30 9.96260 Tf 9.78711 -3.61602 Td [ (to) -223.99200 (a) -224.00700 (manifold\055) ] TJ -180.32200 -11.95510 Td [ (v) 24.98110 (alued) -324.01700 (point) ] TJ /R40 9.96260 Tf 53.25900 0 Td [ (m) -0.30142 ] TJ /R42 9.96260 Tf 13.68790 0 Td [ (2) -416.68300 (M) -0.89936 ] TJ /R30 9.96260 Tf 25.97890 0 Td (i\056e\056\054) Tj /R36 9.96260 Tf 18.08320 0 Td [ (f) -0.59794 ] TJ /R38 9.96260 Tf 10.09100 0 Td [ (\072) -0.80379 ] TJ /R44 9.96260 Tf 6.90781 0 Td [ (R) -0.19605 ] TJ /R46 6.97380 Tf 7.19492 3.61602 Td [ (N) -0.90321 ] TJ /R42 9.96260 Tf 11.69800 -3.61602 Td [ (\041) -416 (M) -0.89691 ] TJ /R30 9.96260 Tf 26.06720 0 Td [ (\054) -342.99700 (using) -325.01400 (a) -323.99000 (neural) ] TJ -172.96800 -11.95470 Td [ (netw) 10.00810 (ork\054) -249.98800 (where) ] TJ /R40 9.96260 Tf 64.35780 0 Td [ (m) -0.30142 ] TJ /R30 9.96260 Tf 12.03830 0 Td [ (is) -249.98500 (the) -249.99000 (desired) -249.98800 (output\056) ] TJ -64.44100 -12.18130 Td [ (That) -324.01200 (is\054) -342.98700 (this) -324 (paper) -323.98800 (pro) 14.98280 (vides) -324.00500 (a) -323.99000 (frame) 25.01050 (w) 10.00320 (ork) -325 (for) -323.98500 (re) 15.00730 (gression) ] TJ -11.95510 -11.95510 Td [ (that) -272.99900 (is) -274.02000 (applicable) -272.99400 (to) -273.00100 (predicting) -274.00300 (manifold\055v) 25.01050 (alued) -273.00300 (data) -273.00800 (and) -273.98900 (at) ] TJ 11.95470 TL T* [ (the) -324.01700 (same) -324.01500 (time) -322.98300 (is) -324.01200 (able) -323.98300 (to) -324.01200 (le) 25.02030 (v) 14.98280 (erage) -323.99500 (the) -322.99500 (po) 24.98600 (wer) -323.99000 (of) -323.98500 (neural) -324.01900 (nets) ] TJ T* [ (for) -218.00500 (feature) -216.99100 (learning\054) -225.00200 (using) -216.99300 (standard) -217.98800 (backpropag) 4.99426 (ation) -217.98300 (for) -218.00300 (un\055) ] TJ T* [ (constrained) -367.99300 (optimization\056) -663.00200 (In) -368.01700 (this) -367.01200 (paper) 39.99330 (\054) -396.98800 (we) -368.01700 (focus) -368.00700 (on) -368.00700 (tw) 10.00810 (o) ] TJ 11.95590 TL T* [ (manifolds) -255.01600 (that) -255.00200 (are) -254.98900 (of) -255.01400 (wide) -255.00600 (interest) -254.99700 (in) -255.00600 (computer) -254.99200 (vision) -254.98200 (\226) -255.01100 (the) ] TJ 11.95510 TL T* [ (h) 4.98446 (ypersphere) -271.00600 (and) -271.00900 (the) -271.00600 (Grassmannian\056) -373.00100 (W) 79.98660 (e) -271.99900 (describe) -271.00900 (the) -271.00400 (appli\055) ] TJ T* [ (cations) -250.01500 (ne) 15.01710 (xt\056) ] TJ /R28 9.96260 Tf 20.56480 TL T* [ (F) 25.02030 (ace) ] TJ /R42 9.96260 Tf 25.03670 0 Td (\041) Tj /R28 9.96260 Tf 15.33400 0 Td [ (Illumination) -539 (Subspace) -538.98800 (as) -538.99800 (r) 17.99210 (egr) 17.99210 (ession) -539.01300 (on) -538.98800 (the) ] TJ -40.37070 -11.95510 Td (Grassmannian\072) Tj /R30 9.96260 Tf 75.83480 0 Td (The) Tj /R34 9.96260 Tf 19.46020 0 Td [ (illumination) -398.02200 (subspace) ] TJ /R30 9.96260 Tf 93.17700 0 Td [ (of) -398.01200 (a) -398.99700 (human) ] TJ -188.47200 -11.95510 Td [ (f) 9.99588 (ace) -304.98100 (is) -304.99600 (a) -305.01300 (popular) -304.98800 (e) 15.01220 (xample) -304.98800 (from) -304.99800 (computer) -304.98300 (vision) -305.01300 (where) -305.01300 (for) ] TJ ET Q Q Q q q 1 1 1 rg /a0 gs 48.40600 786.42200 515.18800 -52.69900 re f q /s5 gs /x6 Do Q q /s7 gs /x8 Do Q q /s9 gs /x10 Do Q q /s11 gs /x12 Do Q Q Q Q q 1 0 0 1 0 0 cm BT /F1 12 Tf 14.40000 TL ET 1 1 1 rg n 270 32 72 14 re f* 0.50000 0.50000 0.50000 rg BT /F2 9 Tf 10.80000 TL ET BT 1 0 0 1 297 35 Tm (1329) Tj T* ET Q endstream endobj 15 0 obj << /Filter /FlateDecode /Resources << /ExtGState << /a0 << /CA 1 /ca 1 >> >> /XObject << /x18 16 0 R >> >> /Length 28 /Group << /Type /Group /S /Transparency /CS /DeviceRGB /I true >> /BBox [ 78 746 96 765 ] /Type /XObject /Subtype /Form >> stream x+O4PH/VЯ0Pp 0 endstream endobj 16 0 obj << /Filter /FlateDecode /Resources 17 0 R /Length 107 /Type /XObject /BBox [ 78 746 96 765 ] /Subtype /Form >> stream xe AC̬wʠ =p,?]%+H-
Jc "82w8VSnGW;"
endstream
endobj
17 0 obj
<<
/ExtGState <<
/a0 <<
/CA 1
/ca 1
>>
>>
>>
endobj
18 0 obj
<<
/Filter /FlateDecode
/Resources <<
/ExtGState <<
/a0 <<
/CA 1
/ca 1
>>
>>
/XObject <<
/x15 19 0 R
>>
>>
/Length 28
/Group <<
/Type /Group
/S /Transparency
/CS /DeviceRGB
/I true
>>
/BBox [ 67 752 84 775 ]
/Type /XObject
/Subtype /Form
>>
stream
x+O4PH/VЯ04Up
0
endstream
endobj
19 0 obj
<<
/Filter /FlateDecode
/Resources 20 0 R
/Length 228
/Type /XObject
/BBox [ 67 752 84 775 ]
/Subtype /Form
>>
stream
xeQKn!s ?FPav6R٪TS.
b];15YyR
{7QL.\:Rv/x9l+L7h%1!}i/AI(kz"U&,YO![R hg{3}4/GyYF:!w}Gn+'xJcO9i뽼_-:`
endstream
endobj
20 0 obj
<<
/ExtGState <<
/a0 <<
/CA 1
/ca 1
>>
>>
>>
endobj
21 0 obj
<<
/Filter /FlateDecode
/Resources <<
/ExtGState <<
/a0 <<
/CA 1
/ca 1
>>
>>
/XObject <<
/x24 22 0 R
>>
>>
/Length 28
/Group <<
/Type /Group
/S /Transparency
/CS /DeviceRGB
/I true
>>
/BBox [ 113 751 499 772 ]
/Type /XObject
/Subtype /Form
>>
stream
x+O4PH/VЯ02Qp
0
endstream
endobj
22 0 obj
<<
/Filter /FlateDecode
/Resources 23 0 R
/Length 56142
/Type /XObject
/BBox [ 113 751 499 772 ]
/Subtype /Form
>>
stream
xtIH4)ޤv[D."M(T{"p>3}9~{u/09v;{y~t}j , _G} y{.~2Mߵ f]@1 0bZ_>{{cOpC9? 6_x&>>;WYoܐ;6%cՇ@t ˳}l]E~}~炘 !wr3dz`x.(9mA]xsG/xI1>9O1\|-E}Qϋ^~IIo!=#OؾCebOHs? ::N3MԾ:MJɩPu6#>$}qdoQOo϶˧a%}7I6N|_煓Bok̋MAwy-0g eƋEyat_$EkEZ$45H%0\7t:╨^61[Z
ˣɀ} 'ٹKoŝ?#ς'է#Da5#g==8ZM|Wo
D&AY/Lk*:빆W;AܬN#O!Ibvq|]f~ly
OlH5;/5Wx;;`EDCDjpH8Nq(#US_'TIh~`.exC;ggL(.t<\T%wܞr˫,Ӯ}yaO}$o]p"qT :01~/_+vrEc蠊{УK9\ z9٬(䇾SY pw?{β+mEs/:c|l{鵓?q+{_@Z`{G