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Hostage Taker reads:
When Hostage Taker enters the battlefield, exile another target creature or artifact until Hostage Taker leaves the battlefield. You may cast that card for as long as it remains exiled, and you may spend mana as though it were mana of any type to cast that spell.
Inspiring Statuary reads:
Nonartifact spells you cast have improvise.
Can I pay {X} to summon the Hostage Taker's exiled permanent using Improvise (tap a artifact permanent to pay {1} of its cost) up to {0} cost.
No. Hostage Taker says mana may be spent as though it were any type. Improvise does not generate mana, and therefore cannot be used to pay for any mana-type specific parts of the cost. Improvise may still be used to pay for the generic mana portion of the cost (if any).
702.125a Improvise is a static ability that functions while the spell with improvise is on the stack. “Improvise” means “For each generic mana in this spell’s total cost, you may tap an untapped artifact you control rather than pay that mana.”
While Hostage Taker enables you to "spend mana as though it were mana of any type to cast the spell" to pay for the non-generic portion of the mana cost, that does not make it a generic mana cost. The mana in your pool is transmutable, while the mana in the cost retains its mana type. It is a case of all thumbs are fingers but not all fingers are thumbs.
Also from the gatherer entries on Improvise: Improvise can’t pay for White, Blue, Black, Red, Green, or Colorless mana symbols in a spell’s total cost.
Do remember that the statue only gives non-artifacts improvise. This includes ones from zones you would not be able to cast them normally from (Like what is the case from the Hostage Taker.)
