N"#$ * 𝑏! 𝑏 ≔ exp %& '!"#(%& '!$% )(* ハッシュ関数の定義 ℎ 𝕩 = 𝑑 ⊕ 𝑖 = 1 𝑥# 𝜋# 𝑚𝑜𝑑 𝑇 ただし、⊕はbitwise XORで、𝜋# は𝜋* = 1を除き⼗分⼤きい素数 ハッシュの衝突対策は⾏わない Fig. 3. Illustration of the multiresolution hash encoding in 2D. (1) for a given input c assign indices to their corners by hashing their integer coordinates. (2) for all resultin vectors from the hash tables \; and (3) linearly interpolate them according to the relat result of each level, as well as auxiliary inputs b 2 R⇢, producing the encoded MLP in gradients are backpropagated through the MLP (5), the concatenation (4), the linear in Table 1. Hash encoding parameters and their typical values. Only the hash table size ) and max. resolution #max need to be tuned to the use case. Parameter Symbol Value Number of levels ! 16 Max. entries per level (hash table size) ) [214, 224 ] Number of feature dimensions per entry 2 Coarsest resolution #min 16 Finest resolution #max [512, 524288] 3 MULTIRESOLUTION HASH ENCODING Given a fully connected neural network<(y; ), we are interested in an encoding of its inputs y = enc(x;\) that improves the approxima- tion quality and training speed across a wide range of applications without incurring a notable performance overhead. Our neural net- work not only has trainable weight parameters , but also trainable encoding parameters \. These are arranged into ! levels, each con- taining up to) feature vectors with dimensionality . Typical values b eac wh grid is 1 into no opt sub num bou W wh larg 12
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