[{"data":1,"prerenderedAt":339},["ShallowReactive",2],{"navigation":3,"\u002Fconcepts\u002Ftransforms\u002Fspherical-normalize":222,"\u002Fconcepts\u002Ftransforms\u002Fspherical-normalize-surround":334},[4,8,168,206,210,214,218],{"title":5,"path":6,"stem":7},"Quickstart","\u002Fquickstart","01.quickstart",{"title":9,"path":10,"stem":11,"children":12},"Concepts","\u002Fconcepts","20.concepts",[13,15,19,23,84,88,92,96,100],{"title":9,"path":10,"stem":14},"20.concepts\u002Findex",{"title":16,"path":17,"stem":18},"Preprocessing Graph","\u002Fconcepts\u002Fppg","20.concepts\u002F2.ppg",{"title":20,"path":21,"stem":22},"Plane decomposition","\u002Fconcepts\u002Fplanes","20.concepts\u002F3.planes",{"title":24,"path":25,"stem":26,"children":27},"Codecs","\u002Fconcepts\u002Fcodecs","20.concepts\u002F4.codecs",[28,32,36,40,44,48,52,56,60,64,68,72,76,80],{"title":29,"path":30,"stem":31},"Arithmetic","\u002Fconcepts\u002Fcodecs\u002Farithmetic","20.concepts\u002Fcodecs\u002Farithmetic",{"title":33,"path":34,"stem":35},"Context Mixing Lite","\u002Fconcepts\u002Fcodecs\u002Fcontext-mixing-lite","20.concepts\u002Fcodecs\u002Fcontext-mixing-lite",{"title":37,"path":38,"stem":39},"FPC","\u002Fconcepts\u002Fcodecs\u002Ffpc","20.concepts\u002Fcodecs\u002Ffpc",{"title":41,"path":42,"stem":43},"Huffman LLM 5-Bit","\u002Fconcepts\u002Fcodecs\u002Fhuff-llm","20.concepts\u002Fcodecs\u002Fhuff-llm",{"title":45,"path":46,"stem":47},"Huffman","\u002Fconcepts\u002Fcodecs\u002Fhuffman","20.concepts\u002Fcodecs\u002Fhuffman",{"title":49,"path":50,"stem":51},"Identity","\u002Fconcepts\u002Fcodecs\u002Fidentity","20.concepts\u002Fcodecs\u002Fidentity",{"title":53,"path":54,"stem":55},"Neural Predictor","\u002Fconcepts\u002Fcodecs\u002Fneural-predictor","20.concepts\u002Fcodecs\u002Fneural-predictor",{"title":57,"path":58,"stem":59},"Order-1 Arithmetic","\u002Fconcepts\u002Fcodecs\u002Forder1-arithmetic","20.concepts\u002Fcodecs\u002Forder1-arithmetic",{"title":61,"path":62,"stem":63},"Order-1 Scale AC","\u002Fconcepts\u002Fcodecs\u002Forder1-scale-ac","20.concepts\u002Fcodecs\u002Forder1-scale-ac",{"title":65,"path":66,"stem":67},"Per-Group Codebook","\u002Fconcepts\u002Fcodecs\u002Fper-group-codebook","20.concepts\u002Fcodecs\u002Fper-group-codebook",{"title":69,"path":70,"stem":71},"rANS","\u002Fconcepts\u002Fcodecs\u002Frans","20.concepts\u002Fcodecs\u002Frans",{"title":73,"path":74,"stem":75},"tANS","\u002Fconcepts\u002Fcodecs\u002Ftans","20.concepts\u002Fcodecs\u002Ftans",{"title":77,"path":78,"stem":79},"Zstd","\u002Fconcepts\u002Fcodecs\u002Fzstd","20.concepts\u002Fcodecs\u002Fzstd",{"title":81,"path":82,"stem":83},"Zstd Dictionary","\u002Fconcepts\u002Fcodecs\u002Fzstd-dict","20.concepts\u002Fcodecs\u002Fzstd-dict",{"title":85,"path":86,"stem":87},"Container format","\u002Fconcepts\u002Fcontainer","20.concepts\u002F5.container",{"title":89,"path":90,"stem":91},"Delta compression","\u002Fconcepts\u002Fdelta","20.concepts\u002F6.delta",{"title":93,"path":94,"stem":95},"Extensibility","\u002Fconcepts\u002Fextensibility","20.concepts\u002F7.extensibility",{"title":24,"path":25,"stem":97,"children":98},"20.concepts\u002Fcodecs\u002Findex",[99],{"title":24,"path":25,"stem":97},{"title":101,"path":102,"stem":103,"children":104},"Transforms","\u002Fconcepts\u002Ftransforms","20.concepts\u002Ftransforms\u002Findex",[105,106,110,114,118,122,126,130,134,140,144,148,152,156,160,164],{"title":101,"path":102,"stem":103},{"title":107,"path":108,"stem":109},"Alpha-Stable Normalize","\u002Fconcepts\u002Ftransforms\u002Falpha-stable-normalize","20.concepts\u002Ftransforms\u002Falpha-stable-normalize",{"title":111,"path":112,"stem":113},"Bit Reorder","\u002Fconcepts\u002Ftransforms\u002Fbit-reorder","20.concepts\u002Ftransforms\u002Fbit-reorder",{"title":115,"path":116,"stem":117},"Block Microscaling Repack","\u002Fconcepts\u002Ftransforms\u002Fblock-microscaling-repack","20.concepts\u002Ftransforms\u002Fblock-microscaling-repack",{"title":119,"path":120,"stem":121},"Burrows-Wheeler","\u002Fconcepts\u002Ftransforms\u002Fburrows-wheeler","20.concepts\u002Ftransforms\u002Fburrows-wheeler",{"title":123,"path":124,"stem":125},"Byte Split & Nibble Split","\u002Fconcepts\u002Ftransforms\u002Fbyte-split","20.concepts\u002Ftransforms\u002Fbyte-split",{"title":127,"path":128,"stem":129},"Concat","\u002Fconcepts\u002Ftransforms\u002Fconcat","20.concepts\u002Ftransforms\u002Fconcat",{"title":131,"path":132,"stem":133},"Delta","\u002Fconcepts\u002Ftransforms\u002Fdelta","20.concepts\u002Ftransforms\u002Fdelta",{"title":135,"path":136,"stem":137,"children":138},"Index Bitwidth Pack","\u002Fconcepts\u002Ftransforms\u002Findex-bitwidth-pack","20.concepts\u002Ftransforms\u002Findex-bitwidth-pack",[139],{"title":135,"path":136,"stem":137},{"title":141,"path":142,"stem":143},"IntDelta","\u002Fconcepts\u002Ftransforms\u002Fint-delta","20.concepts\u002Ftransforms\u002Fint-delta",{"title":145,"path":146,"stem":147},"Mantissa Zero Strip","\u002Fconcepts\u002Ftransforms\u002Fmantissa-zero-strip","20.concepts\u002Ftransforms\u002Fmantissa-zero-strip",{"title":149,"path":150,"stem":151},"Move to Front","\u002Fconcepts\u002Ftransforms\u002Fmove-to-front","20.concepts\u002Ftransforms\u002Fmove-to-front",{"title":153,"path":154,"stem":155},"MxFp4 Deinterleave","\u002Fconcepts\u002Ftransforms\u002Fmxfp4-deinterleave","20.concepts\u002Ftransforms\u002Fmxfp4-deinterleave",{"title":157,"path":158,"stem":159},"Predictor XOR","\u002Fconcepts\u002Ftransforms\u002Fpredictor-xor","20.concepts\u002Ftransforms\u002Fpredictor-xor",{"title":161,"path":162,"stem":163},"Reshape","\u002Fconcepts\u002Ftransforms\u002Freshape","20.concepts\u002Ftransforms\u002Freshape",{"title":165,"path":166,"stem":167},"Spherical Normalize","\u002Fconcepts\u002Ftransforms\u002Fspherical-normalize","20.concepts\u002Ftransforms\u002Fspherical-normalize",{"title":169,"path":170,"stem":171,"children":172},"Guides","\u002Fguides","30.guides\u002F00.index",[173,174,178,182,186,190,194,198,202],{"title":169,"path":170,"stem":171},{"title":175,"path":176,"stem":177},"CLI usage","\u002Fguides\u002Fcli","30.guides\u002F01.cli",{"title":179,"path":180,"stem":181},"Python API","\u002Fguides\u002Fpython-api","30.guides\u002F02.python-api",{"title":183,"path":184,"stem":185},"Writing your first extension","\u002Fguides\u002Fextension-authoring","30.guides\u002F03.extension-authoring",{"title":187,"path":188,"stem":189},"Trust and security","\u002Fguides\u002Ftrust-and-security","30.guides\u002F04.trust-and-security",{"title":191,"path":192,"stem":193},"Random access","\u002Fguides\u002Frandom-access","30.guides\u002F10.random-access",{"title":195,"path":196,"stem":197},"Tuning","\u002Fguides\u002Ftuning","30.guides\u002F11.tuning",{"title":199,"path":200,"stem":201},"Safetensors","\u002Fguides\u002Fsafetensors","30.guides\u002F50.safetensors",{"title":203,"path":204,"stem":205},"Transformers","\u002Fguides\u002Ftransformers","30.guides\u002F51.transformers",{"title":207,"path":208,"stem":209},"API reference","\u002Fapi","40.api",{"title":211,"path":212,"stem":213},"Benchmarks","\u002Fbenchmarks","50.benchmarks",{"title":215,"path":216,"stem":217},"Extensions","\u002Fextensions","60.extensions",{"title":219,"path":220,"stem":221},"Contributing","\u002Fcontributing","90.contributing",{"id":223,"title":165,"body":224,"description":328,"extension":329,"kind":330,"language":330,"links":330,"meta":331,"navigation":330,"path":166,"seo":332,"stem":167,"summary":330,"__hash__":333},"docs\u002F20.concepts\u002Ftransforms\u002Fspherical-normalize.md",{"type":225,"value":226,"toc":320},"minimark",[227,231,240,245,251,254,293,299,303,310,314],[228,229,165],"h1",{"id":230},"spherical-normalize",[232,233,234,235,239],"p",{},"The ",[236,237,238],"code",{},"SphericalNormalize"," transform performs a per-row (radius, direction) reparameterization on 2D float tensors, capturing row-wise correlations while ensuring a bit-exact round-trip via an XOR residual.",[241,242,244],"h2",{"id":243},"theory","Theory",[232,246,247,248,250],{},"Trained weight matrices often exhibit strong row-wise correlation. ",[236,249,238],{}," exploits this by representing each row as a magnitude (radius) and a unit vector (direction).",[232,252,253],{},"The transform splits the input into three planes:",[255,256,257,273,283],"ol",{},[258,259,260,264,265,268,269,272],"li",{},[261,262,263],"strong",{},"Radius",": A single ",[236,266,267],{},"f32"," value per row, representing the Euclidean norm (",[236,270,271],{},"L2",").",[258,274,275,278,279,282],{},[261,276,277],{},"Direction",": The unit vector of the row, stored as ",[236,280,281],{},"bf16"," elements.",[258,284,285,288,289,292],{},[261,286,287],{},"Residual",": A bit-exact XOR difference between the original float bytes and the reconstructed ",[236,290,291],{},"radius * direction"," prediction bytes.",[232,294,295,296,298],{},"This approach is lossless by construction. Float-op inaccuracy only affects the size of the residual, never the correctness of the final output. The ",[236,297,281],{}," quantized direction vectors compress well, and the XOR residual is concentrated around zero.",[241,300,302],{"id":301},"usage","Usage",[232,304,305,306,309],{},"The transform requires the input to have a ",[236,307,308],{},"Rows"," layout.",[241,311,313],{"id":312},"references","References",[315,316,317],"ul",{},[258,318,319],{},"This bit-exact spherical reparameterization (radius and quantized direction vector with exact XOR residual) is an original transform introduced in PTWM for capturing row-wise correlations in deep learning weights.",{"title":321,"searchDepth":322,"depth":322,"links":323},"",3,[324,326,327],{"id":243,"depth":325,"text":244},2,{"id":301,"depth":325,"text":302},{"id":312,"depth":325,"text":313},"Per-row unit-vector reparameterization with an exact XOR residual.","md",null,{},{"title":165,"description":328},"muBHQ2ExXsSgEvool_b05g288FrFJu26cptHJM8K8s4",[335,337],{"title":161,"path":162,"stem":163,"description":336,"children":-1},"Re-tags a plane's layout without modifying its bytes.",{"title":169,"path":170,"stem":171,"description":338,"children":-1},"Task-oriented walkthroughs for compressing, decompressing, and integrating PTWM.",1784796358019]